ENERGY CONDITIONS and SCALAR FIELD COSMOLOGY by SHAWN WESTMORELAND B.S., University of Texas, Austin, 2001 M.A., University of T

ENERGY CONDITIONS AND SCALAR FIELD COSMOLOGY

SHAWN WESTMORELAND

B.S., University of Texas, Austin, 2001 M.A., University of Texas, Austin, 2004 Ph.D., Kansas State University, Manhattan, 2010

A REPORT

submitted in partial fulﬁllment of the requirements for the degree

MASTER OF SCIENCE

Department of Physics College of Arts and Sciences

KANSAS STATE UNIVERSITY Manhattan, Kansas 2013

Approved by:

Major Professor Bharat Ratra Copyright

Shawn Westmoreland

2013 Abstract

In this report, we discuss the four standard energy conditions of General Relativity (null, weak, dominant, and strong) and investigate their cosmological consequences. We note that these energy conditions can be compatible with cosmic acceleration provided that a repulsive cosmological constant exists and the acceleration stays within certain bounds. Scalar fields and dark energy, and their relationships to the energy conditions, are also discussed. Special attention is paid to the 1988 Ratra-Peebles scalar field model, which is notable in that it provides a physical self-consistent framework for the phenomenology of dark energy. Appendix B, which is part of joint-research with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra, reports on the existence of the Ratra-Peebles scalar field tracker solution in a curvature-dominated universe, and discusses the problem of investigating the evolution of long-wavelength inhomogeneities in this solution while taking into account the gravitational back-reaction (in the linear perturbative approximation). Table of Contents

List of Figures vi

Acknowledgements vii

Dedication viii

1 Energy conditions in classical General Relativity1 1.1 Introduction ...... 1 1.2 The weak energy condition ...... 5 1.3 The null energy condition ...... 8 1.4 The dominant energy condition ...... 12 1.5 The strong energy condition ...... 16 1.6 Convergence conditions ...... 18 1.7 The interrelationships between energy conditions ...... 21

2 Cosmology 24 2.1 Introduction ...... 24 2.2 How energy conditions constrain the cosmic fluid ...... 25 2.3 The constituents of the cosmic fluid ...... 28 2.3.1 Matter ...... 30 2.3.2 Radiation ...... 31 2.3.3 The scalar field ...... 31 2.4 Dark energy ...... 37 2.5 On the possible equations of state ...... 38 2.6 The effective cosmological fluid ...... 40 2.7 The Ratra-Peebles scalar field ...... 42 2.8 Conclusion ...... 44

A The Einstein-Hilbert action with boundary terms 50 A.1 Introduction ...... 50 A.2 The cosmological part ...... 51 A.3 The Ricci part ...... 52 A.4 The boundary term ...... 53 A.5 The matter part ...... 54 A.6 Conclusion ...... 54

iv B On the existence and stability of the Ratra-Peebles scalar ﬁeld tracker solution in a curvature-dominated universe 56 B.1 Introduction ...... 57 B.2 A special solution ...... 59 B.3 The attractor property ...... 61 B.4 Gravitational back-reaction ...... 65

Bibliography 73

v List of Figures

1.1 Interrelationships between energy conditions and convergence conditions. Ac- cording to the conventions of the present work, the case Λ < 0 corresponds to a repulsive (de Sitter) cosmological constant and the case Λ > 0 corresponds to an attractive (anti-de Sitter) one...... 22

2.1 For the toy model of Example 2.3.1, we have plotted a(t) (solid blue), φ(t) (dot-dashed red), and w(t) (dashed orange) all on the same graph with time t running along the horizontal axis. We have set c1 = c2 = 0 and c3 = 1/3. . 36 2.2 A list of substances...... 39

vi Acknowledgments

I would like to express my thanks to Bharat Ratra, for his patient support as my research advisor in the Physics Department at Kansas State University. I wish to spotlight Anatoly Pavlov for very special thanks. The content of Appendix B is part of a joint work that he and I were involved with. I am thankful for his suggestions and useful discussions on certain aspects of this work. I would also like to send my warmest thanks to Larry Weaver, Andrew Ivanov, Omer Farooq, Jianxiong “Jason” Li, Data Mania, Raiya Ebini, Pi-Jung Chang, Louis Crane, Alan Guth, Andy Bennett, Khaled Saaidi, Amit Chakrabarti, Michael J. O’Shea, Mathematica, and especially my loving wife Lei Cao for her endlessly dependable support.

vii Dedication

To Omer: You expressed how eager you were to read this thesis. Here it is.

viii Chapter 1

Energy conditions in classical General Relativity

1.1 Introduction

The Einstein field equation describes the relationship between the stress-energy tensor Tµν of matter-fields and the geometrical properties of spacetime. In units where c = G = 1 (our preferred choice of units in the present work), the Einstein field equation reads: 1 R − Rg + Λg = 8πT , (1.1) µν 2 µν µν µν µν where gµν is the metric tensor, Rµν is the Ricci tensor, R := g Rµν, and Λ is the cosmological constant. In AppendixA, the Einstein field equation is derived using a variational technique.

In principle, one can take any metric gµν imaginable (for example, a traversable wormhole discussed below) and — as long as its second partial derivatives exist — plug that metric into the left hand side of (1.1) to produce the stress-energy tensor corresponding to that metric. In this way, exact solutions to (1.1) can easily be constructed, but the stress-energy tensors will not necessarily be physically reasonable. It is therefore useful to impose one or more energy conditions. Energy conditions serve to precisely codify certain ideas about what is physically reasonable. In the present work, we will study four energy conditions that are in standard usage. These are: the weak energy condition (WEC), the null energy condition (NEC), the dominant energy condition (DEC), and the strong energy condition (SEC).

1 Most of the source material for the present chapter comes from Section 4.3 of The Large Scale Structure of Space-time 1 by Hawking and Ellis, and Section 2.1 of A Relativist’s Toolkit 2 by Eric Poisson.

Example 1.1.1 (a traversable wormhole). The following metric, presented here as a line- element in spherical coordinates, describes a traversable wormhole (cf. Morris and Thorne3);

b(r)−1 ds2 = e2γ(r)dt2 − 1 − dr2 − r2 dθ2 + sin2 θdϕ2 , (1.2) r where b(r) and γ(r) are twice diﬀerentiable functions of r subject to certain restrictions:

(1) There is a ﬁnite positive radial coordinate r0 where b(r0) = r0. (This is the wormhole throat.)

(2)1 − b(r)/r ≥ 0 if r0 ≤ r < rh, where rh is positive and possibly inﬁnite.

0 (3) b (r0) < 1.

(4) b(r) → −Λr3/3 as r → ∞.

(5) e2γ(r) → 1 + Λr2/3 as r → ∞.

Items (1) - (3) ensure that the spatial geometry has the shape of a spherically symmetric wormhole, with item (3) ensuring that the two spherical volumes on each side of the wormhole throat are smoothly joined together (cf. Equation (54) in the paper by Morris and Thorne3). Items (4) and (5) ensure that the metric (1.2) corresponds to de Sitter, anti-de Sitter, or Minkowski spacetime in the asymptotic limit of large r depending on whether Λ < 0, Λ > 0, or Λ = 0, respectively. (According to the sign conventions of the present work, de Sitter spacetime has Λ < 0 and anti-de Sitter spacetime has Λ > 0.) When discussing the topic of traversable wormholes, one usually tries to limit the ac- celerations and tidal forces suﬀered by travelers who pass through the wormhole. We are here omitting such details but the interested reader is referred to the 1988 paper by Morris and Thorne.3 Morris-Thorne wormholes are asymptotically ﬂat and horizonless, but since

2 we allow the possibility that Λ 6= 0, we should not insist on asymptotic ﬂatness. Moreover, in the case where Λ < 0, one expects a cosmological horizon at r = rh. For a dedicated discussion on spherically symmetric traversable wormholes in de Sitter spacetime, the reader is referred to the 2003 paper by Lemos et al.4 According to the Einstein ﬁeld equation (1.1), the wormhole (1.2) has a stress-energy tensor Tµν with non-vanishing components given by:

e2γ(r) (Λr2 + b0(r)) T = tt 8πr2 Λr3 + r + (2rγ0(r) + 1) (b(r) − r) T = rr 8πr2 (b(r) − r)) 2r2(r − b(r))γ00(r) + (rγ0(r) + 1) [2r(r − b(r))γ0(r) − rb0(r) + b(r)] − 2Λr3 T = θθ 16πr 2 Tϕϕ = Tθθ sin θ (1.3)

It is often convenient to express the stress-energy tensor in terms of an orthonormal basis

0ˆ 1ˆ 2ˆ 3ˆ at a base point p. In terms of an orthonormal basis (e0ˆ, e1ˆ, e2ˆ, e3ˆ), or its dual (e , e , e , e ), the metric tensor (locally) has the form of the Minkowski metric, and the stress-energy tensor has the form Tµˆνˆ where T0ˆ0ˆ is the energy density. For i, j ∈ {1, 2, 3}, Tîˆj = Tˆjî is the î, ˆj-component of stress: the î-component of the force exerted by matter-fields across a unit surface with normal vector eˆj. For each i ∈ {1, 2, 3}, Tîî is the normal stress in the (spacelike) eî direction (normal stress is pressure when T1ˆ1ˆ = T2ˆ2ˆ = T3ˆ3ˆ). For i 6= j ∈ {1, 2, 3}, Tîˆj is called shear stress. The components T0ˆî = Tî0ˆ, for i ∈ {1, 2, 3}, are (the negatives of) the components of energy flux as measured by an observer at p with 4-velocity e0ˆ (see Misner, Thorne, and Wheeler5 page 138).

Since Tµν is a symmetric second rank tensor, one can always ﬁnd an orthonormal basis

(e0ˆ, e1ˆ, e2ˆ, e3ˆ) at each point, with e0ˆ future-directed, in which Tµˆνˆ has one of the following mathematically possible forms (cf. Hawking and Ellis1 pages 89 - 90, or Bona et al6 and the references therein):

3   ρ 0 0 0     0 p1 0 0     (first Segrètype)   0 0 p2 0    0 0 0 p3       κ + ν 0 0 −ν    0 p1 0 0     , ν = ±1 (second Segrètype)     0 0 p2 0  Tˆˆ Tˆˆ Tˆˆ Tˆˆ  00 01 02 03  −ν 0 0 ν − κ  Tˆˆ Tˆˆ Tˆˆ Tˆˆ    10 11 12 13  =  T2ˆ0ˆ T2ˆ1ˆ T2ˆ2ˆ T2ˆ3ˆ     κ 0 −1 0 Tˆˆ Tˆˆ Tˆˆ Tˆˆ  30 31 32 33   0 p 0 0     (third Segrètype)   −1 0 −κ 1    0 0 1 −κ      0 0 0 ν     0 p1 0 0     , κ2 < 4ν2 (fourth Segrètype)  0 0 p2 0     ν 0 0 −κ

The ﬁrst Segr`etype corresponds to the unique case where the tangent space at each point has an orthonormal basis of eigenvectors of Tµν. As expressed in matrix form above, the

first Segrètype has a timelike eigenvector e0ˆ with corresponding eigenvalue ρ, and for each i ∈ {1, 2, 3} there is a spacelike eigenvector eî with corresponding eigenvalue −pi. Note that the stress-energy tensor of a perfect fluid is of the first Segrètype with p1 = p2 = p3

2 (cf. Poisson page 30). The second Segr`etype has two spacelike eigenvectors e1ˆ and e2ˆ corresponding to eigenvalues −p1 and −p2 respectively, and a double null eigenvector e0ˆ +e3ˆ corresponding to the double eigenvalue κ. Physically, a stress-energy tensor of the second

1 type describes massless radiation propagating in the direction e0ˆ + e3ˆ (Hawking and Ellis page 90). The third Segr`etype has a spacelike eigenvector e1ˆ corresponding to the eigenvalue

−p and a triple null eigenvector e0ˆ + e3ˆ corresponding to a triple eigenvalue κ. The fourth

Segr`etype has two real eigenvectors, e1ˆ and e2ˆ, which are both spacelike and correspond to 2 2 the eigenvalues −p1 and −p2 respectively. The condition κ < 4ν ensures that there are no other real eigenvalues. Hawking and Ellis1 wrote (page 90) that stress-energy tensors of the

4 third and fourth types do not arise in any known physical processes. In fact, by Theorems 1.3.1, 1.3.4, 1.4.1, and 1.5.1 in the present work, it follows that stress-energy tensors of the third and fourth types violate all four energy conditions. Returning to our wormhole example (1.2), the spherical coordinates (t, r, θ, ϕ) correspond to a coordinate basis (et, er, eθ, eϕ) where et = ∂t, er = ∂r, eθ = ∂θ, and eϕ = ∂ϕ.A transformation to an orthonormal dual basis can be reckoned by glancing at (1.2); one can use:

etˆ = eγ(r)et b(r)−1/2 erˆ = 1 − er r eθˆ = reθ

eϕˆ = r sin θeϕ (1.4)

Expressed in terms of (etˆ, erˆ, eθˆ, eϕˆ), the stress-energy tensor of our wormhole has the following non-vanishing components:

Λr2 + b0(r) T = tˆtˆ 8πr2 (2rγ0(r) + 1) (r − b(r)) − Λr3 − r T = rˆrˆ 8πr3 2r2(r − b(r))γ00(r) + (rγ0(r) + 1) [2r(r − b(r))γ0(r) − rb0(r) + b(r)] − 2Λr3 T = T = θˆθˆ ϕˆϕˆ 16πr3 (1.5)

This is a tensor of the ﬁrst Segr`etype.

1.2 The weak energy condition

µ ν µ The weak energy condition (WEC) asserts that Tµνv v ≥ 0 for all timelike vectors v (see Poisson2 page 30, or cf. Hawking and Ellis1 page 89). Since an observer with 4-velocity

µ µ ν v sees the local energy density as being Tµνv v , the WEC prohibits observers from seeing negative energy densities (Hawking and Ellis1 page 89, Poisson2 page 30). Although it may

5 seem reasonable to postulate that the WEC always holds, experiments have shown that it is violated by certain phenomena such as the Casimir eﬀect. However, the current evidence suggests that there are strong limits on how severe such violations can be, globally (e.g., Poisson2 page 32).

µ ν Since the truth-value of the inequality Tµνv v ≥ 0 is unaﬀected if the nonzero vector vµ is replaced by any nonzero scalar multiple of vµ, it follows that the WEC is equivalent

µ ν µ to the statement that Tµνv v ≥ 0 for all normalized future-directed timelike vectors v . The wormhole spacetime of Example 1.1.1 does not satisfy the WEC. In fact, as we show in Section 1.3, it does not even satisfy the null energy condition, which is weaker than the WEC (see Theorem 1.3.1).

Theorem 1.2.1. If Tµˆνˆ is of the first Segrètype, then the WEC is satisfied if and only if

1 2 ρ ≥ 0 and ρ + pi ≥ 0 for each i ∈ {1, 2, 3} (cf. Hawking and Ellis page 90, Poisson pages 30 - 31).

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed, and in which Tµˆνˆ explicitly has the diagonal form of the ﬁrst Segr`etype. The set

FT p of all normalized future-directed timelike vectors at p is given by:

2 2 2 1/2 FT p = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ where a, b, c ∈ R (1.6)

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set

FT p. Since Tµˆνˆ is of the ﬁrst Segr`etype, we have:

µˆ νˆ 2 2 2 2 2 2 Tµˆνˆv v = (1 + a + b + c )ρ + a p1 + b p2 + c p3 (1.7)

The WEC requires that:

2 2 2 2 2 2 (1 + a + b + c )ρ + a p1 + b p2 + c p3 ≥ 0, for all a, b, c ∈ R (1.8)

In the particular case where a = b = c = 0, statement (1.8) implies that ρ ≥ 0. Taking

2 2 b = c = 0, statement (1.8) implies that (1 + a )ρ + a p1 ≥ 0 for all a ∈ R. Diving both sides 2 by a and taking the limit a → ∞ gives ρ + p1 ≥ 0. Similarly, ρ + p2 ≥ 0 and ρ + p3 ≥ 0.

6 µˆ 2 2 2 1/2 To prove the converse, let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FT p and suppose that ρ ≥ 0 and ρ + pi ≥ 0 for each i ∈ {1, 2, 3}. One

2 2 2 2 2 2 has the following four inequalities: ρ ≥ 0, a ρ + a p1 ≥ 0, b ρ + b p2 ≥ 0, and c ρ + c p3 ≥ 0.

2 2 2 2 2 2 When added together, these give (1 + a + b + c )ρ + a p1 + b p2 + c p3 ≥ 0. That is,

µˆ νˆ µˆ Tµˆνˆv v ≥ 0 for any normalized future-directed timelike vector v .

Theorem 1.2.2. If Tµˆνˆ is of the second Segr`etype, then the WEC is satisﬁed only if κ ≥ 0,

ν = +1, and κ + pi + 1 ≥ 0 for each i ∈ {1, 2}.

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed, and in which Tµˆνˆ has the explicit form given by the second Segr`etype.

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set

FT p. Since Tµˆνˆ is of the second Segr`etype, the WEC requires, for all a, b, c ∈ R:

2 2 2 2 2 2 1/2 2 2 (1 + a + b )(κ + ν) + 2c ν − 2νc(1 + a + b + c ) + a p1 + b p2 ≥ 0 (1.9)

2 2 1/2 With a = b = 0, one gets κ + ν 1 + 2c − 2c(1 + c ) ≥ 0 for all c ∈ R. Note that since 1 + 2c2 − 2c(1 + c2)1/2 → 0 as c → +∞, it follows that κ ≥ 0. Furthermore, since 1 + 2c2 − 2c(1 + c2)1/2 → +∞ as c → −∞, it follows that ν ≥ 0; thus ν = +1. Letting

2 2 b = c = 0 in (1.9) gives (1 + a )(κ + ν) + a p1 ≥ 0 for all a ∈ R. Dividing both sides by 2 a and taking the limit as a → ∞ gives κ + ν + p1 ≥ 0. Since ν = +1, we in fact have

κ + p1 + 1 ≥ 0. Similarly, κ + p2 + 1 ≥ 0.

Theorem 1.2.3. If Tµˆνˆ is of the second Segr`etype, then the WEC is satisﬁed if κ ≥ 0,

1 ν = +1, and pi ≥ 0 for each i ∈ {1, 2} (Hawking and Ellis page 90).

µˆ 2 2 2 1/2 Proof. Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of FT p at a point p, where e0ˆ is future-directed and Tµˆνˆ explicitly has the form of the second Segr`e type with respect to the orthonormal basis (e0ˆ, e1ˆ, e2ˆ, e3ˆ). Taking −T0ˆ3ˆ = −T3ˆ0ˆ = ν = +1,

7 the WEC requires:

2 2 2 2 2 2 1/2 2 2 (1 + a + b )(κ + 1) + 2c − 2c(1 + a + b + c ) + a p1 + b p2 ≥ 0 for all a, b, c ∈ R (1.10)

2 2 Since κ ≥ 0, we have (1 + a + b )κ ≥ 0. Since p1 and p2 are nonnegative, we have

2 2 2 2 2 2 2 2 1/2 a p1 + b p2 ≥ 0. Since 1 + a + b + 2c ≥ ±2c(1 + a + b + c ) (square both sides to prove it), we have that 1 + a2 + b2 + 2c2 − 2c(1 + a2 + b2 + c2)1/2 ≥ 0. Adding these inequalities

µˆ νˆ µˆ together gives (1.10), proving that Tµˆνˆv v ≥ 0 if v is a normalized future-directed timelike vector.

If Tµˆνˆ is of either the third or fourth Segr`etype, then the WEC is not satisﬁed (cf. Hawking and Ellis1 page 90). In fact, we show in Theorem 1.3.4 that these types do not even satisfy the null energy condition.

1.3 The null energy condition

µ ν µ The null energy condition (NEC) asserts that Tµνk k ≥ 0 for all null vectors k (see

2 µ ν Poisson page 31). Since the truth-value of Tµνk k ≥ 0 is unaﬀected if the nonzero vector kµ is replaced by any scalar multiple of kµ, it follows that an equivalent formulation of the

µ ν µ NEC is that Tµνk k ≥ 0 for all future-directed null vectors k .

Theorem 1.3.1. The WEC implies the NEC.

Proof. This follows readily from the fact that a null vector can be obtained as the limit of a sequence of timelike vectors (cf. Hawking and Ellis1 page 89, or 95).

Theorem 1.3.2. If Tµˆνˆ is of the first Segrètype, then the NEC is satisfied if and only if

2 ρ + pi ≥ 0 for each i ∈ {1, 2, 3} (cf. Poisson page 31).

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed and Tµˆνˆ explicitly has the form of the ﬁrst Segr`etype. The set FN p of all future-

8 directed null vectors at p is given by:

2 2 2 1/2 FN p = (a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ where a, b, c ∈ R are not all 0

µˆ 2 2 2 1/2 Let k eµˆ = (a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FN p.

Since Tµˆνˆ is of the ﬁrst Segr`etype, we can write:

µˆ νˆ 2 2 2 2 2 2 Tµˆνˆk k = (a + b + c )ρ + a p1 + b p2 + c p3 (1.11)

The NEC requires that:

2 2 2 2 2 2 (a + b + c )ρ + a p1 + b p2 + c p3 ≥ 0, for all a, b, c ∈ R (1.12)

Taking b = c = 0 and a 6= 0, statement (1.12) readily implies that ρ + p1 ≥ 0. Similarly,

ρ + p2 ≥ 0 and ρ + p3 ≥ 0.

µˆ 2 2 For the converse, suppose that ρ + pi ≥ 0 for each i ∈ {1, 2, 3}. Let k eµˆ = (a + b +

2 1/2 c ) e0ˆ +ae1ˆ +be2ˆ +ce3ˆ be an arbitrary element of the set FN p. One has the following three 2 2 2 2 2 2 inequalities: a ρ + a p1 ≥ 0, b ρ + b p2 ≥ 0, and c ρ + c p3 ≥ 0. When added together, these

2 2 2 2 2 2 µˆ νˆ give (a +b +c )ρ+a p1 +b p2 +c p3 ≥ 0, proving that Tµˆνˆk k ≥ 0 for any future-directed null vector kµˆ.

Theorem 1.3.3. If Tµˆνˆ is of the second Segr`etype, then the NEC is satisﬁed if and only if

ν = +1 and κ + pi ≥ 0 for each i ∈ {1, 2}.

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed, and Tµˆνˆ explicitly has the form of the second Segr`etype.

µˆ 2 2 2 1/2 Let k eµˆ = (a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FN p.

If Tµˆνˆ is of the second Segr`etype, then the NEC requires that:

2 2 2 2 2 21/2 2 2 (κ + ν) a + b + 2c ν − 2νc a + b + c + a p1 + b p2 ≥ 0 (1.13)

2 2 1/2 2 Letting a = 1 and b = 0 gives κ + ν 1 + 2c − 2c (1 + c ) + p1 ≥ 0. Since 1 + 2c − 2c(1 + c2)1/2 → +∞ as c → −∞, it follows that ν ≥ 0; this ν = +1. Taking the limit c → +∞, one gets κ + p1 ≥ 0. Similarly, κ + p2 ≥ 0.

9 For the converse, suppose ν = +1, κ + p1 ≥ 0 and κ + p2 ≥ 0. For any a, b, c ∈ R, one 2 2 2 2 2 2 2 2 1/2 has (κ + p1)a ≥ 0, (κ + p2)b ≥ 0, and since a + b + 2c ≥ ±2c(a + b + c ) , one gets ν 1 + 2c2 − 2c (1 + c2)1/2 ≥ 0. Adding these inequalities together gives (1.13), proving

µˆ νˆ µˆ that Tµˆνˆk k ≥ 0 for any future-directed null vector k .

Theorem 1.3.4. If Tµˆνˆ is of either the third or fourth Segr`etype, then the NEC is not satisﬁed.

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- µˆ 2 2 directed, and Tµˆνˆ explicitly has the form of the third Segr`etype. Let k eµˆ = (a + b +

2 1/2 c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FN p.

With Tµˆνˆ of the third Segr`etype, the NEC requires:

2 2 2 21/2 a (κ + p) + 2bc − 2b a + b + c ≥ 0 for all a, b, c ∈ R (1.14)

However, (1.14) cannot be true because 2bc − 2b (a2 + b2 + c2)1/2 → −∞ as b → +∞ if one takes a = c = 0, for example.

Suppose that, with respect to our orthonormal basis, Tµˆνˆ explicitly has the form of the fourth Segr`etype. In this case, the NEC requires:

2 2 21/2 2 2 2 2νc a + b + c + a p1 + b p2 − c κ ≥ 0 for all a, b, c ∈ R (1.15)

However, this also leads to a contradiction. Let a = b = 0 and c = −|ν|/ν (since κ2 < 4ν2, we can be sure that ν is nonzero). Statement (1.15) then gives κ2 ≥ 4ν2, which contradicts the condition that κ2 < 4ν2.

We will now show that the wormhole spacetime described in Example 1.1.1 violates the NEC. Referring to the equations (1.5), we get that the stress-energy tensor of the wormhole is of the ﬁrst Segr`etype, and using the condition that b(r0) = r0, one gets:

0 b (r0) − 1 T0ˆ0ˆ + T1ˆ1ˆ = 2 (1.16) 8πr0

10 0 0 The NEC thus requires b (r0) ≥ 1, but this contradicts the requirement that b (r0) < 1; the third constraint on the metric (1.2). We have shown that a perfectly spherical traversable wormhole violates the NEC. It turns out that even if the wormhole were not perfectly spherical, the NEC would still be violated. As recorded in the 1988 Morris-Thorne3 paper, Don Page noted that Proposition 9.2.8 in Hawking and Ellis1 (page 320) implies that any traversable wormhole (perfectly spherical or not) must violate the NEC. (Although Morris and Thorne discuss the WEC rather than the NEC, Proposition 9.2.8 only uses the NEC.)

Example 1.3.1 (the Vaidya metric). The Vaidya metric can be constructed by taking the Schwarzschild solution in (ingoing) Eddington-Finkelstein coordinates (v, r, θ, ϕ), and letting the mass m be a twice differentiable function of v-time (cf. Poisson2 page 167). This metric describes the spacetime of a spherically symmetric black hole whose mass m(v) varies with time. For a generic cosmological constant, the Vaidya metric reads: 2m(v) Λr2 ds2 = 1 − + dv2 − 2dvdr − r2dθ2 − r2 sin2 θdϕ2 (1.17) r 3 We will show that the NEC is satisfied if and only if m0(v) ≥ 0. This example illustrates the relationship between the NEC and the second law of black hole mechanics, which is a classical theorem essentially stating that the NEC implies that the surface area of a black hole event horizon (perfectly spherical or not) can never decrease (see Proposition 9.2.7 in Hawking and Ellis1 page 318). For the black hole in the present example, a decreasing surface area corresponds to a decreasing m(v). Classically, one probably would have guessed that m(v) should not decrease because nothing can escape from a black hole. On the other hand, thanks to the Hawking effect, one expects black holes to emit radiation. It is interesting to note that the Hawking effect (which is quantum mechanical) can lead to violations of the NEC: a black hole which is hotter than its surroundings and eats nothing will shrink smaller and smaller (and by the second law of black hole mechanics, this violates the NEC). By plugging (1.17) into the Einstein field equation (1.1) one gets that, with respect to the coordinate basis (ev, er, eθ, eϕ), the only nonzero component of the stress-energy tensor

11 is:

m0(v) T = (1.18) vv 4πr2

One can transform to an orthonormal basis (evˆ, erˆ, eθˆ, eϕˆ) via: m(v) Λr2 evˆ = 1 − + ev − er r 6 m(v) Λr2 erˆ = − ev + er r 6 eθˆ = reθ

eϕˆ = r sin θeϕ (1.19)

We ﬁnd that, with respect to the orthonormal (dual) basis (1.19), the nonzero components of the stress-energy tensor are:

m0(v) T = T = T = T = (1.20) vˆvˆ vˆrˆ rˆvˆ rˆrˆ 4πr2

µˆ The set Np of all null vectors k at a point p is given by:

2 2 2 1/2 Np = ±(a + b + c ) evˆ + aerˆ + beθˆ + ceϕˆ where a, b, c ∈ R

So the NEC holds for this spacetime if and only if:

m0(v) T kµˆkνˆ = 2a2 + b2 + c2 ± 2a(a2 + b2 + c2)1/2 ≥ 0 for all a, b, c ∈ µˆνˆ 4πr2 R (1.21)

Since 2a2 + b2 + c2 ≥ ±2a(a2 + b2 + c2)1/2 for all a, b, c ∈ R (square both sides to prove this), it follows from (1.21) that the NEC is satisﬁed if and only if m0(v) ≥ 0.

1.4 The dominant energy condition

We deﬁne the dominant energy condition (DEC) as follows: for any future-directed timelike

µ µ vector v , the DEC requires that Tµνv is neither past-directed nor spacelike. The DEC

12 appears to be formulated slightly diﬀerently elsewhere (compare page 91 of Hawking and Ellis1 with page 32 of Poisson2), but the main idea of the DEC is that the local energy- momentum should always ﬂow from the past to the future. The DEC gets its name from the fact that it requires the energy density to dominate over the pressure (cf. Hawking and Ellis1 page 91). This fact is illustrated by Theorems 1.4.2 and 1.4.3.

µ Note that an equivalent formulation of the DEC is that Tµνv is neither past-directed nor spacelike for all normalized future-directed timelike vectors vµ. The DEC almost reads like a formulation of local causality, but it is interesting to note that if energy-momentum could flow along spacelike vectors, it would not necessarily lead to causality violations.7 We can illustrate this by considering, for simplicity, a two dimensional Minkowski spacetime. There is only one dimension of space (with only two spatial directions: left and right), and there is time. Suppose that, in this imaginary world, there exists a finite wire made of a very strange substance. Light signals passing through the wire from left to right are unaffected and travel along future-directed null vectors, but light signals passing from right to left always pass through the wire at a superluminal speed and travel along spacelike vectors. One cannot use this set-up to send a signal into one’s own past because the superluminal effect only works in one direction.

Theorem 1.4.1. The DEC implies the WEC.

µ µ Proof. Let v be a future-directed timelike vector. The DEC stipulates that uν := Tµνv

µˆ 0ˆ 0ˆ νˆ is not past-directed. Select an orthonormal basis where v eµˆ = v e0ˆ, v > 0, and uνê = 0ˆ 1ˆ νˆ 0ˆ νˆ u0ê + u1ê . Then uνˆv = u0ˆv . Since u is not past-directed, we have u0ˆ ≥ 0 and thus µˆ νˆ νˆ µ ν µ Tµˆνˆv v = uνˆv ≥ 0. We have therefore shown that Tµνv v ≥ 0 if v is a future-directed timelike vector. Since this inequality still holds if vµ is replaced by −vµ, it holds for past- directed timelike vµ as well.

Theorem 1.4.2. If Tµˆνˆ is of the first Segrètype, then the DEC is satisfied if and only if

1 2 ρ ≥ |pi| for each i ∈ {1, 2, 3} (cf. Hawking and Ellis page 91, Poisson page 32).

13 Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed, in which Tµˆνˆ explicitly has the form of the ﬁrst Segr`etype.

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set

FT p. Since Tµˆνˆ is of the ﬁrst Segr`etype, we can write:

µˆ 2 2 2 1/2 Tµˆνˆv eνˆ = (1 + a + b + c ) ρe0ˆ + ap1e1ˆ + bp2e2ˆ + cp3e3ˆ (1.22)

µˆ According to the DEC, Tµˆνˆv is neither past-directed nor spacelike, so:

(1 + a2 + b2 + c2)1/2ρ ≥ 0, and (1.23)

2 2 2 2 2 2 2 2 2 2 ρ + (ρ − p1)a + (ρ − p2)b + (ρ − p3)c ≥ 0, for all a, b, c ∈ R (1.24)

2 2 It readily follows from (1.23) that ρ ≥ 0, and it follows from (1.24) that ρ − p1 ≥ 0. (Note 2 2 that if ρ − p1 < 0, then statement (1.24) would be contradicted by taking b = c = 0 and 2 2 −1/2 2 2 2 2 a > ρ(p1 − ρ ) .) Similarly, ρ − p2 ≥ 0 and ρ − p3 ≥ 0. So ρ ≥ |pi| for each i ∈ {1, 2, 3}. µˆ 2 To prove the converse, suppose that ρ ≥ |pi| for each i ∈ {1, 2, 3}. Let v eµˆ = (1 + a +

2 2 1/2 b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FT p. Since ρ ≥ |pi| ≥ 0, we 2 2 2 1/2 µˆ readily get that (1+a +b +c ) ρ ≥ 0, and so Tµˆνˆv , given by (1.22), is not past-directed.

Moreover, since ρ ≥ |pi| ≥ 0 for each i ∈ {1, 2, 3}, one has the following four inequalities:

2 2 2 2 2 2 2 2 2 2 ρ ≥ 0, (ρ − p1)a ≥ 0, (ρ − p2)b ≥ 0, and (ρ − p3)c ≥ 0. Adding these together, we get 2 2 2 2 2 2 2 2 2 2 µˆ ρ + (ρ − p1)a + (ρ − p2)b + (ρ − p3)c ≥ 0, proving that Tµˆνˆv is not spacelike for any normalized future-directed timelike vector vµˆ.

Theorem 1.4.3. If Tµˆνˆ is of the second Segr`etype, then the DEC is satisﬁed only if κ ≥ 0 and κ + ν > |pi| for each i ∈ {1, 2}.

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis, with e0ˆ future-directed, in which Tµˆνˆ is explicitly of the second Segr`etype.

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set

14 FT p. Since Tµˆνˆ is of the second Segr`etype, one gets:

µˆ 2 2 2 1/2 0ˆ 1ˆ Tµˆνˆv = (κ + ν)(1 + a + b + c ) − νc e + ap1e

2ˆ 2 2 2 1/2 3ˆ +bp2e + (ν − κ)c − ν(1 + a + b + c ) e (1.25)

µˆ The DEC requires Tµˆνˆv to be neither not past-directed nor spacelike:

(κ + ν)(1 + a2 + b2 + c2)1/2 − νc ≥ 0, and (1.26)

2 2 2 1/2 2 2 2 (κ + ν)(1 + a + b + c ) − νc − a p1

2 2 2 2 2 1/22 −b p2 − (ν − κ)c − ν(1 + a + b + c ) ≥ 0 (1.27)

Letting a = b = c = 0 in (1.26), one gets that κ + ν ≥ 0. Letting a = b = 0 and c > 0 in (1.26), dividing both sides of (1.26) by c and taking the limit as c → +∞, one gets that

2 2 2 2 κ ≥ 0. Letting b = c = 0 in (1.27), one gets that (κ +2κν)(1+a )−a p1 ≥ 0. Dividing both 2 2 2 2 2 sides by a 6= 0 and taking the limit as a → ∞, one gets κ + 2κν ≥ p1. Thus, (κ + ν) > p1, and since κ + ν ≥ 0, it follows that κ + ν ≥ |p1|. Similarly, κ + ν ≥ |p2|.

Theorem 1.4.4. The DEC is satisﬁed if κ ≥ 0, ν = +1, and κ ≥ |pi| for each i ∈ {1, 2} (cf. Hawking and Ellis1 page 91).

µˆ µˆ Proof. With κ ≥ 0 and ν = +1, then we have (1.26), so Tµˆνˆv is not past-directed if v is a normalized future-directed timelike vector. With ν = +1, (1.27) reads:

2 2 2 2 2 2 1/2 2 2 2 2 2 2 2 2 2 2κ + 2a κ + 2b κ + 4c κ − 4cκ(1 + a + b + c ) + κ + a κ + b κ − a p1 − b p2 ≥ 0 (1.28)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 With κ ≥ |pi|, we have κ ≥ pi . So κ +a κ +b κ −a p1 −b p2 ≥ 0. Since 1+a +b +2c ≥ 2c(1+a2 +b2 +c2)1/2, and κ ≥ 0, we get 2κ+2a2κ+2b2κ+4c2κ−4cκ(1+a2 +b2 +c2)1/2 ≥ 0.

µˆ Adding all of these inequalities together gives (1.28), proving that Tµˆνˆv is not spacelike if vµˆ is a normalized future-directed timelike vector.

15 1.5 The strong energy condition

1 µ ν The strong energy condition (SEC) asserts that Tµν − 2 T gµν v v ≥ 0 for any timelike vector vµ (Hawking and Ellis1 page 95, or Poisson2 page 31). Note that the SEC does not imply the WEC (Hawking and Ellis1 page 95, Poisson2 page 31, or Section 1.7 in our present work).

1 µ ν Note that an equivalent formulation of the SEC is that Tµν − 2 T gµν v v ≥ 0 for all normalized future-directed timelike vectors vµ.

Theorem 1.5.1. The SEC implies the NEC

Proof. A null vector can be obtained as the limit of a sequence of timelike vectors.

By Theorems 1.5.1, 1.4.1, and 1.3.1, it follows that any violation of the NEC also violates the SEC, WEC, and DEC. Thus, the wormhole spacetime of Example 1.1.1 violates all four energy conditions. Also, the Vaidya spacetime of Example 1.3.1 violates all four energy conditions if m0(v) < 0. Why then is it widely held that a hot black hole shrinking itself away (via the Hawking effect) is physically plausible, but traversable wormholes are not? A good reason is that the Hawking effect can be understood in terms of very plausible physical mechanisms, such as virtual particle-antiparticle annihilation at the event horizon or in terms of particles quantum-tunneling out of the horizon.8 On the other hand, there are presently no plausible mechanisms for the creation of traversable wormholes. Comparing and contrasting traversable wormholes with black hole evaporation illustrates that although violations of the energy conditions may occur in physically implausible scenarios, their violation does not necessarily imply that the scenario is physically implausible. Although the energy conditions are typically obeyed in classical systems, one can encounter exceptions in certain quantum mechanical systems (see also the discussion on page 32 of Poisson2). Dark energy apparently violates the SEC. Indeed, it is easy to construct counterexamples to the SEC using scalar fields. We will say more about this in Chapter2

16 Theorem 1.5.2. If Tµˆνˆ is of the first Segrètype, then the SEC is satisfied if and only if

1 ρ + p1 + p2 + p3 ≥ 0 and ρ + pi ≥ 0 for each i ∈ {1, 2, 3} (cf. Hawking and Ellis page 95, Poisson2 page 31).

Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- directed, in which Tµˆνˆ explicitly has the form of the ﬁrst Segr`etype.

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set

FT p. Since Tµˆνˆ is of the ﬁrst Segr`etype, we have:

T = ρ − p1 − p2 − p3, and (1.29)

µˆ νˆ 2 2 2 2 2 2 Tµˆνˆv v = (1 + a + b + c )ρ + a p1 + b p2 + c p3 (1.30)

1 µˆ νˆ According to the SEC, Tµˆνˆ − 2 T gµˆνˆ v v ≥ 0, so for all a, b, c ∈ R: 1 1 1 1 a2 + b2 + c2 + ρ + a2 + p + b2 + p + c2 + p ≥ 0 (1.31) 2 2 1 2 2 2 3

Taking a = b = c = 0, (1.31) readily gives ρ + p1 + p2 + p3 ≥ 0. Taking b = c = 0, statement

2 2 (1.31) says that (a + 1/2) ρ + (a + 1/2) p1 + p2/2 + p3/2 ≥ 0 for all a. Dividing both sides

2 of this latter inequality by a and taking the limit as a → ∞, one gets ρ + p1 ≥ 0. Similarly,

ρ + p2 ≥ 0 and ρ + p3 ≥ 0.

To prove the converse, suppose that ρ+p1+p2+p3 ≥ 0 and ρ+pi ≥ 0 for each i ∈ {1, 2, 3}.

µˆ 2 2 2 1/2 Let v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FT p. 2 2 One has the following four inequalities: ρ/2 + p1/2 + p2/2 + p3/2 ≥ 0, a ρ + a p1 ≥ 0,

2 2 2 2 2 2 2 b ρ + b p2 ≥ 0, c ρ + c p3 ≥ 0. Adding them all together gives (a + b + c + 1/2) ρ +

2 2 2 (a + 1/2) p1 + (b + 1/2) p2 + (c + 1/2) p3 ≥ 0. Thus, for any normalized future-directed

µˆ 1 µˆ νˆ timelike vector v , we have Tµˆνˆ − 2 T gµˆνˆ v v ≥ 0.

Theorem 1.5.3. If Tµˆνˆ is of the second Segr`etype, then the SEC is satisﬁed only if ν = +1, p1 + p2 ≥ 0, and κ + pi + 1 ≥ 0 for each i ∈ {1, 2}.

17 Proof. Let (e0ˆ, e1ˆ, e2ˆ, e3ˆ) be an orthonormal basis at a base point p, where e0ˆ is future- µˆ 2 2 directed, in which Tµˆνˆ is explicitly of the second Segr`etype. Let v eµˆ = (1 + a + b +

2 1/2 c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ be an arbitrary element of the set FT p. Since Tµˆνˆ is of the second Segr`etype, we have:

T = 2κ − p1 − p2, and (1.32)

µˆ νˆ 2 2 2 2 2 2 1/2 2 2 Tµˆνˆv v = (1 + a + b )(κ + ν) + 2c ν − 2νc(1 + a + b + c ) + a p1 + b p2

(1.33)

1 µˆ νˆ According to the SEC, Tµˆνˆ − 2 T gµˆνˆ v v ≥ 0, so for all a, b, c ∈ R: 1 1 (1 + a2 + b2)(κ + ν) + 2c2ν − 2νc(1 + a2 + b2 + c2)1/2 + a2p + b2p − κ + p + p ≥ 0 1 2 2 1 2 2 (1.34)

2 2 1/2 1 1 2 With a = b = 0, we have ν 1 + 2c − 2c(1 + c ) + 2 p1 + 2 p2 ≥ 0. Since 1 + 2c − 2c(1 + 2 1/2 2 2 1/2 c ) → 0 as c → +∞, it follows that p1 + p2 ≥ 0. Since 1 + 2c − 2c(1 + c ) → +∞ as

2 2 2 c → −∞, it follows that ν = +1. Letting b = c = 0, one gets ν + a ν + a κ + (a + 1/2) p1 +

2 p2/2 ≥ 0. Dividing both sides by a and taking the limit a → ∞ gives ν + κ + p1 ≥ 0. Since we in fact have ν = +1, this means κ + p1 + 1 ≥ 0. Similarly, κ + p2 + 1 ≥ 0.

Theorem 1.5.4. If Tµˆνˆ is of the second Segr`etype, then the SEC is satisﬁed if ν = +1,

1 p1 + p2 ≥ 0, and κ + pi ≥ 0 for each i ∈ {1, 2} (cf. Hawking and Ellis page 95).

Proof. With ν = +1, we have ν + a2ν + b2ν + 2c2ν − 2νc(1 + a2 + b2 + c2)1/2 ≥ 0 for all

2 2 2 2 a, b, c. With κ + p1 ≥ 0 and κ + p2 ≥ 0, we have a κ + a p1 ≥ 0 and b κ + b p2 ≥ 0. With p1 + p2 ≥ 0, we have p1/2 + p2/2 ≥ 0. Adding these results together gives (1.34).

1.6 Convergence conditions

The roots of the energy conditions run deep in classical General Relativity, particularly with respect to the celebrated singularity theorems of Hawking and Penrose. The most important

18 singularity theorems in Hawking and Ellis1 tend to rely on either the timelike convergence condition (TCC) or the null convergence condition (NCC). In fact, the NCC is equivalent to the NEC, and in the special case where Λ = 0, the TCC is equivalent to the SEC (see Theorem 1.6.2). The TCC states that the expansion of a congruence of timelike geodesics (with zero vorticity) monotonically decreases along a timelike geodesic (in other words, according to the TCC, gravity is always attractive), and the NCC says likewise for null congruences. By the Raychaudhuri equation, these conditions can be expressed in terms of the Ricci tensor

µ ν µ Rµν. The TCC requires Rµνv v ≥ 0 for all timelike vectors v , and the NCC requires

µ ν µ 1 Rµνk k ≥ 0 for all null vectors k (see Hawking and Ellis pages 94 - 95).

µ ν Equivalently stated, the TCC requires Rµνv v ≥ 0 for all normalized future-directed

µ µ ν timelike vectors v and the NCC requires Rµνk k ≥ 0 for all future-directed null vectors kµ.

Theorem 1.6.1. The TCC implies the NCC.

Proof. A null vector can always be obtained as the limit of a sequence of timelike vectors.

Theorem 1.6.2. If Λ = 0, then the TCC is equivalent to the SEC. If Λ > 0, then the SEC implies the TCC. If Λ < 0, then the TCC implies the SEC. Moreover, the NCC is equivalent to the NEC (independent of the value of Λ).

Proof. By index gymnastics, the Einstein ﬁeld equation (1.1) can be put into the form:

1 R − Λg = 8π T − T g (1.35) µν µν µν 2 µν

By (1.35), the SEC is equivalent to the following:

µ ν µ (Rµν − Λgµν) v v ≥ 0 for all timelike vectors v (1.36)

This reduces to the TCC when Λ = 0. Similarly, it follows from (1.35) that the NEC is equivalent to the NCC, independent of the value of Λ.

19 µ ν µ ν µ By (1.36), the SEC states that Rµνv v ≥ Λgµνv v for all timelike vectors v . (Hence, according to the SEC, if gravity can be repulsive then it must be that Λ < 0.) Since

µ ν gµνv v > 0 for timelike vectors, it follows that the SEC implies the TCC if Λ > 0.

µ ν µ The TCC states that Rµνv v ≥ 0 for all timelike vectors v . So, if Λ < 0, the TCC

µ ν µ ν µ implies that Rµνv v ≥ Λgµνv v for all timelike vectors v . Hence, the TCC implies the SEC if Λ < 0.

Theorem 1.6.3. If Tµˆνˆ is of the first Segrètype, then the TCC is satisfied if and only if

1 ρ + p1 + p2 + p3 + Λ/(4π) ≥ 0 and ρ + pi ≥ 0 for each i ∈ {1, 2, 3} (cf. Hawking and Ellis page 95).

1 Λ µˆ νˆ Proof. By (1.35), the TCC requires that Tµˆνˆ − 2 T gµˆνˆ + 8π gµˆνˆ v v ≥ 0 for all timelike µˆ µˆ v . With Tµˆνˆ of the ﬁrst Segr`etype, and v ∈ FT p a normalized future-directed timelike vector, this means for all a, b, c ∈ R: 1 1 1 1 Λ a2 + b2 + c2 + ρ + a2 + p + b2 + p + c2 + p + ≥ 0 2 2 1 2 2 2 3 8π (1.37)

Λ Setting a = b = c = 0, one gets ρ + p1 + p2 + p3 + 4π ≥ 0. Taking b = c = 0, we have 2 2 Λ 2 (a + 1/2)ρ + (a + 1/2)p1 + p2/2 + p3/2 + 8π ≥ 0. Dividing both sides by a and taking the limit as a → ∞ gives ρ + p1 ≥ 0. Similarly, ρ + p2 ≥ 0 and ρ + p3 ≥ 0.

Λ 2 2 2 2 For the converse, write ρ/2 + p1/2 + p2/2 + p3/2 + 8π ≥ 0, a ρ + a p1 ≥ 0, b ρ + b p2 ≥ 0, 2 2 µˆ νˆ c ρ + c p3 ≥ 0, and add the inequalities together to get (1.37). That is, Rµˆνˆv v ≥ 0 for all normalized future-directed timelike vectors vµˆ.

Theorem 1.6.4. If Tµˆνˆ is of the second Segr`etype, then the TCC is satisﬁed only if κ ≥ −1,

ν = +1, and p1 + p2 + Λ/(4π) ≥ 0.

Proof. For the second type, TCC requires, for normalized future-directed timelike vectors

20 µˆ v ∈ FT p:

(a2 + b2)(κ + ν) + ν 1 + 2c2 − 2c(1 + a2 + b2 + c2)1/2 1 1 Λ + a2 + p + b2 + p + ≥ 0 2 1 2 2 8π (1.38)

2 2 1/2 Λ Letting a = b = 0 gives ν 1 + 2c − 2c(1 + c ) + p1/2 + p2/2 + 8π ≥ 0. Since we 2 2 1/2 Λ have 1 + 2c − 2c(1 + c ) → 0 as c → ∞, it follows that p1 + p2 + 4π ≥ 0. Since 1 + 2c2 − 2c(1 + c2)1/2 → +∞ as c → −∞, it follows that ν = +1. Letting b = c = 0,

2 Λ 2 gives a (κ + ν) + ν + p1/2 + p2/2 + 8π ≥ 0. Dividing both sides by a and taking the limit as a → ∞, one gets κ + ν ≥ 0. Since ν = +1, this means κ ≥ −1.

Theorem 1.6.5. If Tµˆνˆ is of the second Segr`etype, then the TCC is satisﬁed if κ ≥ −1,

1 ν = +1, p1 + p2 + Λ/(4π) ≥ 0 and pi ≥ 0 for each i ∈ {1, 2} (cf. Hawking and Ellis page 95).

Proof. With ν = +1 and κ ≥ −1, we have (a2+b2)(κ+ν) ≥ 0 and ν 1 + 2c2 − 2c(1 + c2)1/2 ≥

2 2 0. With p1, p2 ≥ 0 we have a p1 ≥ 0 and b p2 ≥ 0. With p1 + p2 + Λ/(4π) ≥ 0, we have p1/2 + p2/2 + Λ/(8π) ≥ 0. Adding all these together gives (1.38).

Since the TCC implies the NCC (= NEC) it follows by Theorem 1.3.4 that the TCC does not hold if Tµν is of either the third or fourth Segr`etype.

1.7 The interrelationships between energy conditions

Theorems 1.3.1, 1.4.1, 1.5.1, 1.6.1, and 1.6.2 can all be summarized with a single diagram (see Figure 1.1). The purpose of this section is to show that the diagram of Figure 1.1 is complete in the sense that any implication which cannot be inferred from the diagram is false. To this end, a minimal set of three counterexamples can be used. One expects these counterexamples to be

21 Λ<0

yÑ Λ=0 DEC SEC ks +3 TCC :B Λ>0

& WEC +3 NEC ks +3 NCC

Figure 1.1: Interrelationships between energy conditions and convergence conditions. Ac- cording to the conventions of the present work, the case Λ < 0 corresponds to a repulsive (de Sitter) cosmological constant and the case Λ > 0 corresponds to an attractive (anti-de Sitter) one. somewhat exotic and immodest because the energy conditions, by design, require physical tameness and normalcy. It is possible to describe each counterexample as a peculiar kind of perfect fluid. Recall that a perfect fluid has a stress-energy tensor of the first Segrètype which, when expressed in terms of an orthonormal basis of eigenvectors, has ρ := T0ˆ0ˆ 6= 0 and p := T1ˆ1ˆ = T2ˆ2ˆ = T3ˆ3ˆ. A perfect fluid has an equation of state parameter w given by w := p/ρ. Perfect fluids and their applications to cosmology will be revisited in Section 2.2. The first of our three counterexamples is dark energy, which is a perfect fluid with −1 ≤ w < −1/3. (In fact, for reasons spelled out in Section 2.4, dark energy will be defined so that it has −1 ≤ w < −1/3 and ρ > 0, but for the present section — and the present section only — we are ignoring the ρ > 0 requirement.) If ρ > 0, then dark energy satisfies DEC, and consequently the WEC and the NEC, but it does not satisfy the SEC. If w = −1 and ρ < 0, then dark energy satisfies the SEC, and the consequently the NEC, but not the WEC and not the DEC. With ρ < 0 and w > −1, dark energy violates the NEC, and consequently violates all the other conditions as well. Note that dark energy satisfies the TCC if Λ > 0 and 0 < ρ ≤ −Λ/(4π(1 + 3w)). Dark energy with w = −1 and ρ > |Λ|/(8π) violates the TCC. Dark energy with w = −1 and ρ < −|Λ|/(8π) satisfies the TCC, but not

22 the WEC and not the DEC. The second counterexample is phantom quintessence, or phantom energy,9 which has w < −1. If ρ > 0, then phantom quintessence violates the NEC. If ρ < 0, then phantom quintessence satisfies the SEC and consequently the NEC, but not the WEC and not the DEC. Note that phantom quintessence with ρ < 0 violates the TCC if Λ < 0 and −Λ/(4π(1+ 3w)) < ρ < 0. The third counterexample is a hypothetical substance that we name phantom quaint- essence. We define this as a strange type of perfect fluid with w > 1. If ρ > 0, then phantom quaint-essence satisfies the WEC, SEC, and the NEC, but not the DEC. If ρ < 0, then phantom quaint-essence violates the NEC. Phantom quaint-essence and phantom quintessence share the common feature that they both violate the DEC. Our neologism ‘quaint-essence’ is a play on the word ‘quintessence,’ which is another name for dark energy. Of course, ‘phantom quaint-essence’ is a parody of ‘phantom quintessence’ or ‘phantom energy.’ We will have more to say about quaint-essence, dark energy, and phantom quintessence in Chapter2. For the present, it suffices to note that dark energy with ρ > 0 can be used to get that NEC ; SEC, WEC ; SEC, and DEC ; SEC. Dark energy with ρ > |Λ|/(8π) and w = −1 gives DEC ; TCC, WEC ; TCC, and NEC ; TCC. Dark energy with ρ < 0 and w = −1 gives NEC ; WEC and NEC ; DEC. To get that TCC ; WEC and TCC ; DEC, consider dark energy with w = −1 and ρ < −|Λ|/(8π). Moreover, dark energy with 0 < ρ ≤ −Λ/(4π(1 + 3w)) gives TCC ; SEC if Λ > 0. By considering phantom quintessence with ρ < 0, one gets that SEC ; WEC and SEC ; DEC. Phantom quintessence with −Λ/(4π(1 + 3w)) < ρ < 0 can be used to show that SEC ; TCC if Λ < 0. Finally, one can use phantom quaint-essence with ρ > 0 to get that WEC ; DEC.

23 Chapter 2

Cosmology

2.1 Introduction

The main purpose of the present Chapter is to discuss some elementary concepts in cosmology from the perspective of energy conditions. In Section 2.2, the energy conditions will be applied to Friedmann cosmology as a whole and certain inequalities are derived. It will be noted that some energy conditions predict upper-bounds on the recently discovered phenomenon of cosmic acceleration. The idea of using the energy conditions to constrain cosmology has previously been developed by Visser.10 Our present study is in a spirit similar to Visser’s,10 and we would have simply repeated parts of his analysis almost exactly except for one detail which may be significant. Like Visser, we begin with the Friedmann metric, compute the corresponding stress-energy tensor, and then impose constraints on this stress-energy tensor using the energy conditions. The difference between our analysis and Visser’s is that we compute the stress-energy tensor using the Einstein field equation with the cosmological constant, whereas Visser evidently used the Einstein field equation without the cosmological constant (or with Λ = 0). This difference appears to have consequences. For example, we get the result — in contrast to Visser10 — that the SEC does not necessarily prohibit the Universe from accelerating. Let us mention that some cosmologists have used observational data to check whether or

24 not energy conditions are violated in the Universe. To name a few papers in this category, let us mention Sen and Scherrer,11 Schuecker et al.,12 Qiu, Cai, and Zhang,13 Wu, Ma, and Zhang,14 and Lima, Vitenti, Rebou¸cas.15 These studies are consistent with the previously mentioned program of Visser’s10 and so they are based on a philosophy that is slightly different from ours. In the present Chapter, we will also consider the idea of a cosmic scalar field φ, and how energy conditions constrain the potential of the scalar field in a very elementary way. The Ratra-Peebles scalar field and the flat φCDM model is discussed in Section 2.7. Further work, relating to the extension of the original φCDM model to non-flat cosmologies (part of joint-work with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra16), is discussed in AppendixB.

2.2 How energy conditions constrain the cosmic ﬂuid

The cosmological principle asserts that the Universe is, on the average, homogeneous and isotropic. Put another way, the Universe is (approximately) spherically symmetric about every point. Due to a 1944 theorem of A. G. Walker, it turns out that the only type of geometry with (exact) spherical symmetry about every point is the geometry given by the Friedmann metric (Hawking and Ellis1 page 135):

dr2 ds2 = dt2 − a(t)2 + r2dθ2 + r2 sin2 θdϕ2 . (2.1) 1 − K2r2

The scale factor a(t) is a nonnegative twice diﬀerentiable function with continuous derivatives. Spacelike hypersurfaces of constant t are homogeneous isotropic 3-spaces of constant curvature. These 3-spaces are hyperbolic, ﬂat, or hyperspherical depending on whether K2 is negative, zero, or positive, respectively.

We can transform from the coordinate basis (et, er, eθ, eϕ) to an orthonormal basis

25 (etˆ, erˆ, eθˆ, eϕˆ) via:

etˆ = et a(t) erˆ = er (1 − K2r2)1/2 eθˆ = a(t)reθ

eϕˆ = a(t)r sin θeϕ (2.2)

Using the Einstein ﬁeld equation (1.1), we ﬁnd that, with respect to the orthonormal basis

(2.2), the nonzero components of the stress-energy tensor Tµˆνˆ corresponding to the metric (2.1) are:

1 3K2 3˙a(t)2 T = Λ + + tˆtˆ 8π a(t)2 a(t)2

1 K2 a˙(t)2 2¨a(t) T = T = T = − Λ + + + (2.3) rˆrˆ θˆθˆ ϕˆϕˆ 8π a(t)2 a(t)2 a(t)

Therefore, in cosmology, the total stress-energy tensor has the form of a perfect fluid, with energy density ρ = Ttˆtˆ and pressure p = Trˆrˆ. We are apt to call this the cosmic fluid. Let us consider how the energy conditions constrain the cosmic fluid. Of all the energy conditions studied in Chapter1, the least restrictive is the null energy condition (NEC). For the cosmic fluid, Theorem 1.3.2 assures that the NEC is satisfied if and only if:

K2 +a ˙(t)2 a¨(t) ≤ (2.4) a(t)

The phenomenon of cosmic acceleration, which is actually observed in our current epoch, corresponds to havinga ¨(t) > 0. Note that the NEC puts an upper bound ona ¨(t). Following Visser,10 let us remark that the NEC is related to the idea that if the Universe expands its density ought to decrease, and if the Universe contracts its density ought to

µν increase. To get this, note that local energy conservation ∇µT = 0 implies that:

ρ˙ = −3(ρ + p)˙a(t)/a(t) (2.5)

26 The NEC requires ρ + p ≥ 0. Note that, by (2.5), ρ + p > 0 if and only ifρ ˙ anda ˙(t) have opposite signs. By Theorem 1.2.1, the cosmic fluid satisfies the weak energy condition (WEC) if and only if it satisfies the NEC and:

3K2 3˙a(t)2 Λ + + ≥ 0 (2.6) a(t)2 a(t)2

That is, the WEC requires that the energy density of the cosmic fluid must be positive.10 Note that this introduces certain restrictions on Λ and K2. In the present work, the most restrictive energy conditions that we have studied are the dominant (DEC) and strong (SEC) energy conditions. For the cosmic fluid, it follows from Theorem 1.4.2 that the DEC is satisfied if and only if:

K2 a˙(t)2 2K2 2˙a(t)2 a¨(t) + − a¨(t) Λ + + + ≥ 0 (2.7) a(t) a(t) a(t)2 a(t)2 a(t)

Since the DEC implies the NEC, and the NEC requiresa ¨(t) ≤ K2/a(t) +a ˙(t)2/a(t), it follows that — as long as we do not have the special casea ¨(t) = K2/a(t)+a ˙(t)2/a(t) (which obtains if and only if ρ + p = 0) — one can infer the following constraint from the DEC:

2K2 2˙a(t)2 a¨(t) Λ + + + ≥ 0 (2.8) a(t)2 a(t)2 a(t)

Put another way, the DEC introduces a lower bound ona ¨(t):

2K2 2˙a(t)2 a¨(t) ≥ −Λa(t) − − (2.9) a(t) a(t)

Theorem 1.5.2 assures that the SEC is satisﬁed if and only if (2.4) holds and:

Λa(t) a¨(t) ≤ − (2.10) 3

Thus, the SEC limits cosmic acceleration more strongly than the NEC and permitsa ¨(t) > 0 only if the cosmological constant acts repulsively (Λ < 0). The relation (2.10) can also be derived from the Raychaudhuri equation.1

1Cf. lecture notes by Hirata, 17 but note that our Λ apparently has the opposite sign of his. There are also slight diﬀerences between our views and his regarding the relationship between Λ and the SEC.

27 We note that Theorem 1.6.3 assures that the cosmic ﬂuid satisﬁes the timelike convergence condition (TCC) if and only if (2.4) holds and:

a¨(t) ≤ 0 (2.11)

Hence, the TCC is incompatible with cosmic acceleration. The inequalities that we have here derived are not necessarily identical to those derived by Visser10 nor by Cattoënand Visser.18 Note that the cosmic fluid in the other references corresponds to ours only in the special case where Λ = 0. (Evidently, in the other references, the stress-energy tensor of the cosmic fluid was obtained from the Einstein field equation with the cosmological constant set equal to zero.) Consequently, the other references get for example that the SEC requiresa ¨(t) ≤ 0. We remark that Visser10 also used the energy conditions to derive constraints on the Hubble parameter. Cattoënand Visser18 went further to derive constraints on luminosity and angular-size distances, look-back time, and other cosmological parameters. In Sec- tion 2.8, we will rewrite (2.4), (2.6), (2.9), (2.10), and (2.11) in terms of the redshift, the deceleration parameter, the Hubble parameter, etcetera.

2.3 The constituents of the cosmic ﬂuid

Letting ρ and p denote the total energy density and pressure of the cosmic ﬂuid, one can rewrite equations (2.3) as:

3˙a(t)2 3K2 = 8πρ − Λ − a(t)2 a(t)2

a˙(t)2 2¨a(t) K2 + = −8πp − Λ − (2.12) a(t)2 a(t) a(t)2

These are the Friedmann equations. In fact, the equation in the second line of (2.12) can be obtained by taking the derivative of the equation in ﬁrst line with respect to t, while taking into account the fact that according to the ﬁeld equation (1.1), energy is locally conserved

28 µν 2 2 ∇µT = 0. We remark that, by using the two equations in (2.12), one can eliminate K and write: a¨ 4π Λ = − (ρ + 3p) − (2.13) a 3 3 It is customary to regard the cosmological constant Λ as having an effective energy density ρΛ and pressure pΛ such that Λ ρ = −p = − (2.14) Λ Λ 8π That is, the cosmological constant can be thought of as a perfect fluid with equation of state parameter w = −1. Similarly, the curvature of the spacelike hypersurfaces of constant t has an effective energy density ρK and pressure pK given by: 3K2 ρ = −3p = − (2.15) K K 8πa(t)2 That is, curvature can be thought of as a perfect fluid with w = −1/3. Using (2.14) and (2.15), one can rewrite the Friedmann equations (2.12) as: 3˙a(t)2 = 8π (ρ + ρ + ρ ) a(t)2 Λ K

a˙(t)2 2ä(t) + = −8π (p + p + p ) (2.16) a(t)2 a(t) Λ K The total energy density ρ and pressure p of the cosmic fluid comes from matter, radiation, plus any other auxiliary fields. In the present study, we will be concerned with only one auxiliary field: a scalar field φ. Letting ρM , ρR and ρφ denote respectively the energy densities of matter, radiation, and the φ-field, and letting pM , pR and pφ denote respectively the pressures of matter, radiation, and the φ-field, one has:

ρ = ρM + ρR + ρφ

p = pM + pR + pφ (2.17)

2 µν The local conservation law ∇µT = 0 follows from the Einstein field equation thanks to the purely µν µν µν mathematical identities ∇µg = 0 and ∇µR = ∇µ (Rg /2), though one could point out that ensuring µν ∇µT = 0 was always one of the motivations behind the field equation in the first place.

29 2.3.1 Matter

In cosmology, the total matter content in the Universe is taken to be a monoenergetic gas of particles all moving with the same speed, but in random directions. The stress-energy tensor for a monoenergetic gas can be understood in the following way (cf. Martin19 pages 124 - 126). First, consider dust. A dust is a continuous collection of particles all moving with the same speed and direction. For a dust with proper energy density ρ0 and 4-velocity

µˆ 2 2 2 1/2 v eµˆ = (1 + a + b + c ) e0ˆ + ae1ˆ + be2ˆ + ce3ˆ, in an orthonormal basis (e0ˆ, e1ˆ, e2ˆ, e3ˆ), the stress-energy tensor Tµˆνˆ is:  1 a b c  2 ρ0  a a ab ac  Tµˆνˆ = ρ0vµˆvνˆ =   (2.18) 1 − v2  b ba b2 bc  c ca cb c2 where v2 = a2 +b2 +c2 is the square of the 3-dimensional speed of the dust. A monoenergetic gas can be regarded as the average of a very large number of dusts, all moving with 3- dimensional speed v, but in random directions. In this way, the oﬀ-diagonal terms in

Tµˆνˆ for the gas will average out to zero, and the squares of the three orthogonal velocity components (a2, b2, and c2) all average out to be v2/3. Therefore, the stress-energy tensor of a monoenergetic gas (or the total cosmological matter) is of the form:  1 0 0 0  2 ρ0  0 v /3 0 0  Tµˆνˆ =   (2.19) 1 − v2  0 0 v2/3 0  0 0 0 v2/3 2 2 This is a perfect ﬂuid with energy density ρ = ρ0/(1 − v ) and pressure p = ρv /3. The equation of state is therefore given by w = v2/3. Since, for massive particles, 0 ≤ v2 < 1, it follows that 0 ≤ w < 1/3. In cosmology, the energy density contributions from matter are overwhelmingly in the form of cold dark matter (the average speed v of the matter particles is much less than the speed of light). Hence, to a good approximation, the equation of state for the total cosmological matter is w = 0, and the pressure due to matter is negligible.

30 2.3.2 Radiation

Radiation, regarded as a part of the cosmic ﬂuid, can be thought of as the limiting case of a monoenergetic gas when the particles become massless and travel at the speed of light. By taking the limit v2 → 1 in (2.19), the stress-energy tensor for radiation, with energy density ρ, is found to have the following form (in an orthonormal basis):

 1 0 0 0   0 1/3 0 0  Tµˆνˆ = ρ   (2.20)  0 0 1/3 0  0 0 0 1/3

Note that the pressure is p = ρ/3, and therefore radiation has the equation of state w = 1/3.

2.3.3 The scalar ﬁeld

Proposals involving scalar fields in gravity and cosmology are nothing new. The 1961 theory of Brans and Dicke20 proposed to implement Mach’s Principle by replacing the gravitational constant with a scalar field. As noted by Hawking and Ellis1 (page 59), the Brans-Dicke theory is equivalent to ordinary General Relativity (with Λ = 0) supplemented by an auxiliary scalar field. Cosmic scalar fields also appeared in Hoyle and Narlikar’s21 work on steady state cosmology, and in Dirac’s22 cosmological work. In most models dealing with early-Universe inflationary scenarios, the inflation effect is driven by a scalar field called the inflaton. Scalar fields have also been used to model dark energy, which is a hypothetical type of energy that behaves similarly to a cosmological constant, causing the expansion of the Universe to accelerate in the present epoch. Of particular importance is the 1988 proposal of Peebles and Ratra,23 in which the cosmological constant becomes, in a sense, a dynamic quantity. Although the Ratra-Peebles proposal is currently used as a model for dark energy, it is interesting to note that it actually predates the experimental discovery of cosmic acceleration by a decade. In accordance with the formalism developed in AppendixA, the simplest action for

31 General Relativity with a scalar field φ is of the form: Z Z 1 µν 1 1 λ S = dv g ∂µφ∂νφ − V (φ) + L − (R − 2Λ) + dΣλB + S0, D 2 16π 16π ∂D (2.21) where L is the matter Lagrangian density for all matter-fields expect for the φ-field. We are assuming that the scalar field φ couples neither to curvature (in accordance with the strong equivalence principle) nor to any other matter-fields.3 The effects of the φ-field can be felt by the other matter-fields only indirectly through gravity. The form of V (φ) will vary from proposal to proposal. In the 1988 Ratra-Peebles23 φCDM proposal, which will be discussed in more detail in Section 2.7 (and extended somewhat in AppendixB), it is postulated that Λ = 0 and V (φ) = βφ−α, where α and β are constants which must be determined empirically.4 In fact, the constants α and β are interrelated (cf. equation (6) in Peebles and Ratra23). By varying the metric inside D and keeping it fixed on ∂D (following methods discussed in detail in AppendixA), the field equations according to (2.21) read:

1 R − Rg + Λg = 8π (T + Q ) , (2.22) µν 2 µν µν µν µν where Tµν is the stress-energy tensor for all matter ﬁelds except for φ, and Qµν is the stress-energy tensor for the φ-ﬁeld:

1 Q = ∂ φ∂ φ − gζξ∂ φ∂ φ − V (φ) g (2.23) µν µ ν 2 ζ ξ µν

The equation of motion for the φ-ﬁeld itself reads:

µν 0 ∇µ (g ∂νφ) + V (φ) = 0 (2.24)

3For this reason, our present formalism is probably too simple to provide a useful framework for early- Universe inflationary scenarios. The hypothetical inflaton field responsible for driving inflation coupled nontrivially to the other matter-fields. 24 However, it is not unreasonable to postulate that the interactions between the inflaton and the other matter-fields became negligible (and remained so) as soon as the early- Universe inflation stopped and reheating commenced. Indeed, the present work is chiefly concerned with late-time cosmology rather than early-Universe cosmology. 4The notation used by Ratra and Peebles, in their action and in their potential, is slightly different from the notation used in the present Chapter.

32 Specializing (2.24) to the Friedmann metric (2.1), and taking (in accordance with the cosmological principle) the φ-ﬁeld to be a function of t only, one gets that, with respect to the coordinate basis (et, er, eθ, eϕ), the nonzero components of Qµν are:

1 Q = φ˙2 + V (φ) (2.25) tt 2

1 ˙2 2 2 φ − V (φ) a(t) Q = (2.26) rr 1 − K2r2

1 Q = φ˙2 − V (φ) r2a(t)2 (2.27) θθ 2

1 Q = φ˙2 − V (φ) r2a(t)2 sin2 θ (2.28) ϕϕ 2

We can transform to an orthonormal basis (etˆ, erˆ, eθˆ, eϕˆ) using (2.2). In this basis, the stress-energy tensor for the scalar ﬁeld has the form:

 1 ˙2  2 φ + V (φ) 0 0 0 1 ˙2  0 2 φ − V (φ) 0 0  Qµˆνˆ =  1 ˙2  (2.29)  0 0 2 φ − V (φ) 0  1 ˙2 0 0 0 2 φ − V (φ) Thus, the stress-energy tensor of the φ-ﬁeld has the form of a perfect ﬂuid with energy density ρφ and pressure pφ given by:

1 ρ = φ˙2 + V (φ) (2.30) φ 2 and

1 p = φ˙2 − V (φ) (2.31) φ 2

We remark that although the stress-energy of a cosmological scalar field has the form of a perfect fluid, perturbations in a scalar field do not act like perturbations in an ordinary perfect fluid.25

33 By plugging the Friedmann metric (2.1) into the ﬁeld equations (2.22), one obtains the following version of the Friedmann equation:

3˙a(t)2 = 8π (ρ + ρ + ρ + ρ + ρ ) , (2.32) a(t)2 M R φ Λ K and equation (2.24) reads:

a˙(t) φ¨ + 3 φ˙ + V 0(φ) = 0 (2.33) a(t)

By (2.30) and (2.31), we get that the equation of state for the φ-ﬁeld is:

1 φ˙2 − V (φ) w = 2 (2.34) 1 ˙2 2 φ + V (φ) In contrast to matter, which has 0 ≤ w < 1/3, the value of w for scalar fields is not restricted. Consider a real Klein-Gordon field φ with mass m. The potential is V (φ) = m2φ2/2. Assuming this potential, equation (2.34) implies that w ≤ 1 if m2 ≥ 0, with w = 1 in the massless case m = 0. In the tachyonic case m2 < 0, one gets |w| > 1. In Section 1.7, we considered hypothetical substances with equations of state w < −1 and w > 1, called phantom quintessence and phantom quaint-essence, respectively. We note that a tachyonic Klein-Gordon field with φ˙2 + m2φ2 < 0, if such exists, would be a type of phantom quintessence, and a tachyonic Klein-Gordon field with φ˙2 +m2φ2 > 0, if such exists, would be a type of phantom quaint-essence. In the next example, we construct a scalar field with a time-dependent equation of state where w ranges over all the values between 1 and −1.

Example 2.3.1. [a toy model of a scalar-ﬁeld dominated cosmology] Let ρM = ρR = ρΛ =

ρK = 0, and let V (φ) = 1/2. Then equations (2.32) and (2.33) read: a˙(t)2 4π = φ˙2 + 1 (2.35) a(t)2 3 and

3˙a(t) φ¨ + φ˙ = 0 (2.36) a(t)

34 A large class of solutions (but not the complete general solution) to the system of equations

(2.35) and (2.36) is as follows, where c1, c2, and c3 are arbitrary constants: √ 1 −1 φ(t) = √ tanh exp −2t 3π + c1 + c2 (2.37) 3π and

1 √ 3 a(t) = c3 sinh 2t 3π − c1 (2.38)

In this model, we get from (2.34) that w is a function of time given by: √ 2 w(t) = 2sech 2t 3π − c1 − 1, (2.39) so −1 < w(t) ≤ 1. Figure 2.1 shows the time evolution of a(t), φ(t), and w(t) with c1 = c2 = 0 and c3 = 1/3.

In the toy model plotted in Figure 2.1, there is a big bang singularity at t = 0. (Indeed, it is truly singular: one can verify that the Kretschmann invariant diverges there.) Although we do not seriously propose that this toy model is an accurate representation of the physical Universe, it has some superficial parallels with the standard picture of cosmology. Namely, in the early stages of the universe, the scalar field equation of state for our toy model evolves from being radiation-like (w ≈ 1/3) to being matter-like (w ≈ 0), and evolves into a dark energy state (w ≈ −1) at late times, resulting in a cosmic acceleration where a(t) grows exponentially. One should not read too much into this. Moreover, unlike the standard cosmological picture, there is an era with 1/3 < w ≤ 1 in the early universe for our toy model. The following theorem describes the extent to which the various energy conditions restrict the scalar field potential V (φ), in a scalar field-dominated universe.

Theorem 2.3.1. Consider the stress-energy tensor of a homogenous and isotropic real scalar

field φ with energy density ρφ and pressure pφ given by (2.30) and (2.31). Then the φ-field satisfies:

35 2.0

1.5

1.0

0.5

t 0.2 0.4 0.6 0.8 1.0

-0.5

-1.0

Figure 2.1: For the toy model of Example 2.3.1, we have plotted a(t) (solid blue), φ(t) (dot-dashed red), and w(t) (dashed orange) all on the same graph with time t running along the horizontal axis. We have set c1 = c2 = 0 and c3 = 1/3.

• the NEC for any V (φ)

1 ˙2 • the WEC if and only if V (φ) ≥ − 2 φ

• the DEC if and only if V (φ) ≥ 0

• the SEC if and only if V (φ) ≤ φ˙2

Proof. By Theorem 1.3.2, the φ-field satisfies the NEC if and only if ρφ + pφ ≥ 0. By ˙2 equations (2.30) and (2.31), one gets ρφ + pφ = φ . Insofar as φ is a real scalar field, we will always have φ˙2 ≥ 0. Thus, the NEC will be satisfied regardless of how V (φ) is defined. By Theorem 1.2.1, the φ-field satisfies the WEC if and only if it satisfies the NEC and 1 ˙2 ρφ ≥ 0. By (2.30), the condition ρφ ≥ 0 is equivalent to V (φ) ≥ − 2 φ .

36 By Theorem 1.4.2, the φ-ﬁeld satisﬁes the DEC if and only if ρφ ≥ |pφ|. By equations

(2.30) and (2.31), one gets ρφ ≥ |pφ| if and only if V (φ) ≥ 0.

By Theorem 1.5.2, the φ-ﬁeld satisﬁes the SEC if and only if ρφ +3pφ ≥ 0 and ρφ +pφ ≥ 0.

As we have seen, the inequality ρφ + pφ ≥ 0 holds regardless. By (2.30) and (2.31), one gets ˙2 ρφ + 3pφ ≥ 0 if and only if V (φ) ≤ φ .

Note that, by Theorem 2.3.1, if the φ-ﬁeld violates the SEC, then it automatically satisﬁes the DEC and WEC.

2.4 Dark energy

At the end of Section 2.2, it was noted that the phenomenon of cosmic acceleration violates the TCC. By Theorem 1.6.2, it follows that if the TCC is violated then there are exactly two possibilities: either 1.) Λ < 0 (i.e., we have a repulsive cosmological constant) or 2.) the SEC is violated. These two possibilities are not necessarily mutually exclusive, but we take it that the main idea of dark energy, or quintessence, is to explain the cosmic acceleration in terms of a dynamical agent instead of a fixed constant such as Λ. Dark energy therefore must violate the SEC. At this point it would not be unreasonable to formally define dark energy as a hypothetical substance that violates the SEC (and presently dominates the Universe). However, we shall distinguish between dark energy proper, and an extreme form of dark energy known as phantom energy or phantom quintessence. Phantom quintessence, a hypothetical substance with equation of state w < −1, could lead to a Big Rip doomsday singularity.26 However, as noted in Section 1.7, phantom quintessence with positive energy density violates the NEC (and therefore all the other energy conditions), while phantom quintessence with negative energy density satisfies the NEC and the SEC, but violates both the WEC and the DEC. Neither dark energy nor phantom quintessence abide by the energy conditions very well, but phantom quintessence is definitely the more delinquent of the two. In order to make a clear distinction between phantom and dark energy proper, we are

37 apt to formally define dark energy as a hypothetical substance which violates the SEC but satisfies the WEC. With this definition, dark energy has an equation of state such that:

1 − 1 ≤ w < − (2.40) 3

As we have seen in Subsection 2.3.3, a scalar ﬁeld can give rise to equation of state like (2.40). Indeed, recall that the scalar ﬁeld equation of state (2.34) reads:

1 φ˙2 − V (φ) w = 2 (2.41) 1 ˙2 2 φ + V (φ)

In the special case where φ˙ = 0 and V (φ) 6= 0, one has the equation of state w = −1, which is the equation of state that corresponds to the cosmological constant. Indeed, when a scalar field has φ˙2 2V (φ), it behaves very similarly to a cosmological constant, which is why scalar fields are often used in models of dark energy. If we define dark energy as a substance that violates the SEC (but satisfies the WEC), then it follows by Theorem 2.3.1 that a real scalar field φ gives rise to some type of dark energy if and only if V (φ) > φ˙2.

2.5 On the possible equations of state

The equation of state for matter has 0 ≤ w < 1/3, and radiation has w = 1/3. Dark energy, or (ordinary) quintessence, has −1 ≤ w < −1/3, and phantom energy, or phantom quintessence, has w < −1. We used the idea of phantom quaint-essence, which has w > 1, in Section 1.7. As mentioned previously, our neologism ‘quaint-essence’ is meant to mirror the word ‘quintessence,’ so to this end we give the name quaint-essence to any quaint sort of substance with 1/3 < w ≤ 1. Work from Subsection 2.3.3 shows that quaint-essence can arise from scalar fields. Thus far, we have discussed substances with w-values ranging over the entire real number line except for the interval −1/3 ≤ w < 0 (though curvature effectively has w = −1/3). Let us define weak quintessence to be a hypothetical substance with −1/3 ≤ w ≤ 0. Both weak

38 quintessence and ordinary quintessence have w < 0, but weak quintessence cannot cause cosmic acceleration. Moreover, we note that weak quintessence with positive energy density satisfies all four energy conditions. Figure 2.2 summarizes our list of various hypothetical substances. Concerning the hypothetical substance quaint-essence, which has never been observed, we remark that we are not the first to consider the possibility that such a thing might exist. In a 1961 paper on the structure of matter-fields at ultrahigh densities, Zel’dovich27 theorized that equations of state with 1/3 < w ≤ 1 can arise when a massive vector field interacts with point charges. Also, the scalar field of the Brans-Dicke theory has pressure equal to its energy density,28 and is therefore a case of quaint-essence with w = 1.

substance eqn. of state comments phantom quintessence w < −1 Big Rip, violates WEC (non-phantom) dark energy/quintessence −1 ≤ w < −1/3 68% of the Universe weak quintessence −1/3 ≤ w < 0 e.g. curvature (cold) matter w = 0 32% of the Universe (hot) matter 0 < w < 1/3 not signiﬁcant today radiation w = 1/3 dominated early quaint-essence 1/3 < w ≤ 1 ultrahigh densities? phantom quaint-essence w > 1 violates DEC

Figure 2.2: A list of substances.

Of all the substances listed in Figure 2.2, only two are signiﬁcant in the present cosmological epoch. These are: 1.) cold matter (this include both luminous and dark matter), and 2.) dark energy. According to results announced by the Planck Collaboration in 2013,29 the so-called energy budget of the physical Universe at the present time is about 32% cold matter and 68% dark energy.

39 2.6 The eﬀective cosmological ﬂuid

In the present work, we have been careful to define the cosmic fluid as the stress-energy tensor given by (2.3). The cosmic fluid consists of matter, radiation, and possibly a scalar field. We do not include the cosmological constant Λ nor do we include curvature as part of the cosmic fluid proper. However, as discussed in Section 2.3, it is a common practice among cosmologists to look on Λ and curvature as having equations of state w = −1 and w = −1/3, respectively. Cosmologists have the notion that the energy-budget of the Universe goes beyond just matter and radiation — it includes the cosmological constant and curvature (if there is any).

To this end, let us define the effective cosmic fluid as having energy density ρe = ρ+ρΛ +

ρK and pressure pe = p + pΛ + pK , where ρ and p denotes the energy density and pressure of the (proper) cosmic fluid defined by (2.3), and ρΛ, ρK , pΛ, pK , denote the effective energy densities and pressures associated with Λ and curvature — see equations (2.14) and (2.15). The concept of the effective cosmic fluid allows us to write the Friedmann equations in the following very compact form:

3˙a(t)2 = 8πρ (2.42) a(t)2 e

a˙(t)2 2¨a(t) + = −8πp (2.43) a(t)2 a(t) e By taking the time-derivative of (2.42), one gets:

2 3˙a(t)¨a(t) = 4π ρ˙ea(t) + 2ρea(t)˙a(t) (2.44)

Solving (2.43) fora ¨(t) and substituting into (2.44) gives:

3˙a(t) a˙(t)2 −4πp a(t)2 − = 4πρ˙ a(t)2 + 8πρ a(t)˙a(t) (2.45) a(t) e 2 e e

2 2 By (2.42), we havea ˙(t) = 8πρea(t) /3. Hence, equation (2.45) can be written as: 3˙a(t) 4πρ a(t)2 −4πp a(t)2 − e = 4πρ˙ a(t)2 + 8πρ a(t)˙a(t) (2.46) a(t) e 3 e e

40 A bit of algebra leads to the eﬀective ﬂuid equation: a˙(t) ρ˙ = −3(ρ + p ) (2.47) e e e a(t) We remark that an apparently alternative way to get (2.47) would be to consider a perfect

µν fluid with energy density ρe and pressure pe and then apply the conservation law ∇µT = 0. However, although such a method would give the correct equation, it is inconsistent with the formalism developed in the present paper; the effective cosmological fluid is not given by the stress-energy tensor for spacetime. The stress-energy tensor for (the Friedmann) spacetime is dictated by the Einstein field equation, and this is how we defined the (proper) cosmic fluid. As the Universe evolves, it goes through phases where the effective cosmic fluid is dominated by one particular type of substance (e.g., matter or radiation) for a given time (i.e., an epoch). Defining the effective equation of state parameter for the Universe as we := pe/ρe, we can write (2.47) as:

ρ˙e a˙(t) = −3 (1 + we) (2.48) ρe a(t) For example, the matter-dominated epoch of the Universe obtains while one has the situation

ρM ρR, ρK , ρΛ, etcetera. The parameter we maintains the value 2/3 throughout the entire duration of the matter-dominated epoch.

Integrating (2.48), and assuming that we 6= −1 is constant with respect to t, one gets:

−3(1+we) ρe = ρ0a(t) , (2.49) where ρ0 is a constant. Substituting (2.49) into (2.42) gives: 8πρ a˙(t)2 = 0 a(t)−1−3we (2.50) 3

Integrating (2.50) and remembering that we is approximately constant throughout the epoch leads to:

n 2 a(t) = a0 (t − t0) , where n = , and a0, t0 are constants. (2.51) 3(1 + we)

41 Thus, when a substance with equation of state we 6= −1 dominates for the duration of a cosmological epoch, the cosmic scale factor follows a power law.5 For example, during a matter-dominated epoch, one has we = 0, and so n = 2/3. During a radiation-dominated epoch, one has we = 1/3, and so n = 1/2. Since the eﬀective energy density of curvature (2.15) is positive only for K2 < 0, a curvature-dominated epoch can occur only in the case K2 < 0 (i.e., in the case of a hyperbolic spatial geometry). For a curvature-dominated epoch one has we = −1/3 and therefore n = 1.

The special case we = −1 occurs, for example, if we have a Λ-dominated epoch. In such a scenario, one gets that a(t) is an exponential function of t.

2.7 The Ratra-Peebles scalar ﬁeld

In 1988, Ratra and Peebles proposed what has come to be known as the φCDM model. This model was developed in order to solve two problems. The first problem is the so-called coincidence problem, and the second is an energy problem. The coincidence problem, as it is normally stated today, is the puzzling observation that we are currently living in a cosmological epoch where the effective energy density of Λ (which remains constant for all time) just happens to be comparable to the energy density of (cold dark) matter.30 Ratra and Peebles introduced their model a decade before the existence of dark energy was firmly established, but already by the mid-1980’s it was evidently apparent that a cosmological constant Λ might be needed in order to reconcile the observed value of the density parameter with the observed flatness of the Universe.23 However, it was considered puzzling that the required Λ and the cold dark matter contributed to the expansion rate nearly equally.23 As for the energy problem, this has to do with the fact that the required Λ corresponds to an

23 energy EΛ given by: 1 3(1 − Ω)H 2 3c5 4 E = 0 ~ , (2.52) Λ 8πG

5 Normally in the literature, a0 denotes the scale factor of the Universe at the current time, but this convention is not always followed in the present paper. We hope that it will always be clear from context whether or not this convention is followed.

42 2 where Ω := 8πGρM /(3H0 ), H0 is the present value of the Hubble parameter H :=a/a ˙ , ~ is the reduced Planck constant, c is the speed of light, and G is the gravitational constant.

−1 −1 Using Ω = 0.32, and H0 = 67.5 km s Mpc (in order to be consistent with the latest

29 results from the Planck Collaboration ), we get EΛ = 2.2 meV. It was argued that it would be unnatural to introduce a new energy scale so small ∼ 10−3 eV.25 In an effort to overcome these problems, Ratra and Peebles,23 kept the formal assumption that Λ = 0 and proposed that a scalar field φ (identified as the remnant of the inflaton field which drove the inflationary phase of the very early Universe) affects a Λ-like term whose energy density decreases asymptotically to zero as it evolves over time. This is achieved by postulating that the potential of the φ-field is of the form

V (φ) = βφ−α, (2.53) where α and β are positive constants. (In AppendixB, the actual formalism used by Ratra and Peebles is discussed in more detail.)6

The energy-density ρφ corresponding the choice of potential (2.53) is, by (2.30):

1 ρ = φ˙2 + βφ−α (2.54) φ 2

In fact, Ratra and Peebles included certain peculiar factors in their action (see equation (B.1) in AppendixB), making it slightly diﬀerent from (2.21), and in their notation the density of their scalar ﬁeld actually reads:

m 2 ρ = p Φ˙ 2 + κm 2φ−α , (2.55) Φ 32π p where mp denotes the Planck mass and κ is a constant analogous to our β in (2.53). (We denote the scalar field as Φ (capital Greek letter) when working in the notation of Ratra and Peebles.) In order to ensure that nucleosynthesis proceeds unaffected by the φ-field in the early Universe, Ratra and Peebles postulated that ρΦ ρ at early times (early times

6The program of Ratra and Peebles succeeds in solving the energy problem only for suﬃciently large α 23 (e.g., α & 6). For details see Section IV in Peebles and Ratra.

43 that are nevertheless post-reheating). Assuming a potential V (φ) of the form (2.53), and a(t) ∼ tn, a stable (attractor) solution to the scalar ﬁeld equation of motion (2.33) was found such that:23

4 ρ a(t) n(α+2) Φ = , (2.56) ρM a1

Note that, by equation (2.56), the Φ-ﬁeld energy density ρΦ decreases less rapidly than the matter density ρM , and moreover there is a special epoch a1 at which the Φ-ﬁeld begins

23 23 to dominate over cold dark matter. As Ratra and Peebles noted, a1 should closely correspond to the present epoch. We remark that the Ratra-Peebles scalar field satisfies the NEC, the WEC, and the DEC. The SEC must be violated if the Ratra-Peebles field is supposed to model dark energy. Ratra and his students have, among other things, used observational data to put constraints on α. The latest results seem to indicate that α is very close to zero (for details, see e.g., Farooq and Ratra31). However, the previous studies assumed (flat) φCDM as the fiducial model, which may have influenced the results. If one adds a new parameter to the model, such as curvature, then there is a possibility that the data will favor a nonzero value for α. The present author is, at the time of this writing, involved in a joint project (with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra16) which seeks to extend the original φCDM model in order to include curvature. Some of this joint work is discussed in AppendixB. At the time of this writing, the joint paper is in-progress, but nearly finished.

2.8 Conclusion

Our discussion from Section 2.2 shows that the four energy conditions (NEC, WEC, DEC, and SEC) can all be consistent with cosmic acceleration provided that there is a repulsive cosmological constant and the cosmic acceleration stays within certain bounds. (However, as noted in Section 2.2, the TCC is incompatible with cosmic acceleration.) To summarize,

44 we found that the SEC constrains cosmic acceleration as follows:

Λa(t) K2 +a ˙(t)2 a¨(t) ≤ min − , (2.57) 3 a(t)

The DEC requires, assuming that we do not have the very special case ρ + p = 0:

2K2 2˙a(t)2 a¨(t) ≥ −Λa(t) − − (2.58) a(t) a(t)

The WEC corresponds to two constraints:

K2 +a ˙(t)2 a¨(t) ≤ , (2.59) a(t) and:

3K2 3˙a(t)2 Λ + + ≥ 0 (2.60) a(t)2 a(t)2

The NEC corresponds to only:

K2 +a ˙(t)2 a¨(t) ≤ (2.61) a(t)

The result that cosmic acceleration does not violate the SEC apparently contradicts results derived by other authors (e.g., Visser10) but the discrepancy arises from the fact that in the present work, the cosmic fluid is defined somewhat differently than it is elsewhere (e.g., compare (2.3) in the present report with equations (3) and (4) in Visser’s10 paper). Following Lima, Vitenti, and Rebou¸cas,15 we remark that in terms of the redshift z :=

2 (a0/a) − 1, the deceleration parameter q(z) := −aa/¨ a˙ , the normalized Hubble function

2 2 E(z) := H(z)/H0, and the curvature density parameter ΩK0 := −K /(a0H0) , the relation (2.61) can be written as:

Ω (1 + z)2 q(z) ≥ K0 − 1 (2.62) E(z)2

2 Deﬁning ΩΛ0 := −Λ/(3H0 ), the restrictiona ¨(t) ≤ −Λa(t)/3 can be written as: Ω q(z) ≥ − Λ0 (2.63) E(z)2

45 Hence, the SEC corresponds to the constraint:

Ω Ω (1 + z)2 q(z) ≥ max − Λ0 , K0 − 1 (2.64) E(z)2 E(z)2

We can rewrite (2.60) as:

E(z)2 − Ω Λ0 ≥ Ω , (2.65) (1 + z)2 K0 and the relation (2.58) can be written as:

2Ω (1 + z)2 + 3Ω q(z) + K0 Λ0 ≤ 2 (2.66) E(z)2

If we set Λ = 0, then our relations (2.63), (2.65), and (2.66) correspond to (24), (23), and (25) in the paper by Lima, Vitenti, and Rebou¸cas.15 By (2.11), the TCC requires:

q(z) ≥ 0, (2.67) which is contradicted by current observations. Many cosmologists believe that a repulsive cosmological constant is a good explanation for the observed cosmic acceleration. Let us remark that Lovelock’s theorem (see Strau- mann32 page 75) appears to provide a good theoretical reason for having the cosmological constant. Lovelock’s theorem (in four dimensions)7 states that if we postulate a ﬁeld equation of the form Aµν[g] = Tµν, where the tensor Aµν[g] is constructed from gµν together with its ﬁrst and second derivatives, then in order to ensure local energy conservation

µν µν (∇µA [g] = ∇µT = 0), it follows that Aµν[g] must be of the form:

Aµν[g] = aGµν + bgµν, (2.68) where a, b ∈ R and Gµν := Rµν − Rgµν/2 is the Einstein tensor. Hence, the Einstein ﬁeld equation with the cosmological constant — which, with units chosen so that G = c = 1, corresponds to setting a = 1/(8π) and b = Λ/(8π) in (2.68) — represents the most general

7There is also a higher dimensional version of Lovelock’s theorem (see Straumann 32 pages 108 - 109).

46 choice of field equation for General Relativity. In other words, the cosmological constant Λ arises naturally from the basic mathematical structure of General Relativity.8 Nevertheless, some cosmologists seek to explain the current cosmic acceleration not strictly in terms of the constant Λ, but in terms of a time-dependent dark energy. We have done our best in Section 2.4 to formulate precisely and fairly what the idea of dark energy means. The coincidence problem and a certain kind of energy problem associated with Λ have already been mentioned in Section 2.7. There is also the issue, frequently raised, that one can attempt to calculate the cosmological constant using quantum field theory, and that the answer obtained through such methods is incorrect by an extremely large order of magnitude. We do not wish to down- play the significance of this problem by any means, but let us note that it identifies the cosmological constant with vacuum energy. Although this identification is widely accepted by physicists, the present author has a somewhat divergent view on this issue. Einstein once described his field equation9 as like a building half made of marble and half made of wood (see Out of My Later Years 34 page 83). The marble half corresponds to the left-hand or geometric side of (1.1), while the wooden half corresponds to the right-hand side of (1.1). By Lovelock’s theorem, the marble half of Einstein’s equation is uniquely determined (in four dimensions), up to constant factors, and the cosmological term belongs there. It is by all means reasonable to use our knowledge of quantum field theory to inform ourselves about

Tµν, the wooden half of Einstein’s equation, but the scope of ordinary quantum field theory is limited. Since ordinary quantum field theory is not a theory of gravity,10 the geometric or marble half of Einstein’s equation is beyond its reach. Quantum field theory may tell us that the vacuum has a stress-energy tensor proportional to gµν (and therefore it mimics a cosmological term), but it remains a separate fact that the cosmological term proper is a property of gravity which is already codified within the marble or geometric half of the

8Weinberg, 33 in his way, also argues that Λ is a natural part of General Relativity. 9Einstein was in fact referring to the version of his ﬁeld equations without Λ, but the present author believes that the version (1.1) with Λ can be described in exactly the same way. 10We do not consider gravitons to be part of ordinary quantum ﬁeld theory.

47 Einstein field equation. We then evidently have two cosmological terms — one from General Relativity and one from quantum field theory — and one can ask if together they produce the effective cosmological term that we observe.11 We do not propose that one is forever stuck with a plurality of fundamentally different cosmological terms (which is theoretically unsatisfying); the situation appears to be a symptom of the fact that the unified theory remains to be understood. As explained in Section 2.4, the dark energy hypothesis (as formulated in the present work) necessarily violates the SEC on the cosmic scale, but it is not at all unprecedented in cosmology to postulate violations of the SEC. For example, the inflationary scenario proposed by Guth36 postulates that an SEC-violating inflaton field briefly dominated shortly after the Big Bang and caused the very early Universe to experience exponential expansion (cf. page 395 of Peebles37).12 Guth introduced the inflationary scenario in order to solve two problems with the non-inflationary Big Bang model. The first being the horizon problem. The horizon problem is the puzzling fact that two regions of the Universe (which could not have been casually connected in the non-inflationary Big Bang picture) are nevertheless observed to be at the same temperature. By introducing SEC violation in the early Universe, the Big Bang singularity is effectively pushed back, so that all regions of the Universe can (at least in principle37) find causal connections to each other in the very distant past. The second problem that inflation aimed to solve was the flatness problem — why does the Universe appear to be so close to being flat? This is especially puzzling given the fact that in order for the Universe to be as flat as it appears today, it must have been extremely fine-tuned to near perfect flatness in the early Universe.36 In the inflationary scenario, the Universe grew at an exponential rate during the short-lived inflationary epoch while maintaining

11Along such lines we may indeed have more than two cosmological terms if there also exists, for example, a dark energy scalar field. Let us remark that Carroll, 35 in Section 1.3 of his review article on the cosmological constant, makes essentially the same point as ours. Furthermore, Carroll points out that one would expect the present-day effective cosmological term to include contributions from electroweak symmetry breaking, chiral symmetry breaking, plus contributions from other early-Universe phase transitions yet to be understood. We are being somewhat simplistic by describing the problem along the lines of only two different cosmological terms, but we hope this is not misleadingly simplistic. 12Exponential expansion in the early Universe was also proposed by Kazanas 38 and Sato 39 independently.

48 an approximately constant energy density, and thereby it decreased the magnitude of its density of curvature (if this was not already zero) by many order of magnitude before the inflationary epoch rapidly died down and reheating commenced. The inflationary proposal takes advantage of the fact that the SEC is easily violated in particle physics. As we saw in Section 2.3.3, it is not difficult to violate the SEC with scalar fields. For a particularly physical example, consider the π0 meson. This is by no means the particle thought to be responsible for inflation, but the existence of π0 mesons is well- established and they worth using in order to make a point. The π0 meson can be described classically as a scalar field satisfying the Klein-Gordon equation (Mandl and Shaw40 page 44) with mass m = 2.4 × 10−25 gm (cf. Particle Data Group41). This corresponds to a Compton wavelength 9.2 × 10−13 cm. Hawking and Ellis1 (page 96) note that although the stress-energy tensor for π0 mesons can violate the SEC, such violations would be localized to distances smaller than 10−12 cm. This might lead one to reconsider the singularity theorems for spacetime when the radius of curvature is less than 10−12 cm, but Hawking and Ellis point out that “such a curvature would be so extreme that it might well count as a singularity” (Hawking and Ellis1 page 96). An intriguing hypothesis is that dark energy is in fact the residual piece of the inflationary field, and we are now entering a new inflationary phase of the Universe. This is essentially the picture coming from the Ratra-Peebles φCDM model, which aims to investigate the late-time consequences of a sterile inflaton field with a decaying power-law potential. The idea is that, although the energy density of the inflation field decays over time, it does so more slowly than the energy density of matter. In this way, the inflation field once again dominates the Universe,23 leading to the accelerated expansion observed today. The beauty of this hypothesis is that it gives a unified explanation for cosmology at late and early times. Careful observations will tell us whether or not the idea is correct, but let us note that the latest efforts of the Planck Collaboration29 did not find any evidence for a dynamical dark energy which apparently strengthens the case for a constant Λ.

49 Appendix A

The Einstein-Hilbert action with boundary terms

A.1 Introduction

Over a four-dimensional compact region D with boundary ∂D, we take the Einstein-Hilbert action with boundary terms to be the action S defined by the following equation Z Z 1 1 λ S := dv L − (R − 2Λ) + dΣλB + S0, (A.1) D 16π 16π ∂D where R is the Ricci scalar curvature, L is the Lagrangian density for the matter-fields, and Λ is the cosmological constant. The Einstein field equation is to be obtained by varying this action with respect to the cometric tensor gµν such that δgµν = 0 on ∂D. The constant

µ term S0 does not aﬀect the equations of motion. In spacetime coordinates x (µ = 0, 1, 2, 3) √ in D, the volume element dv is given by d4x −g, where g is the determinant of the metric

n tensor. In hypersurface coordinates y (n = 1, 2, 3) on ∂D, the surface element dΣλ can be

3 ∂xα ∂xβ ∂xγ 2 expressed as d y ελαβγ ∂y1 ∂y2 ∂y3 (see page 64 in Poisson ), where ελαβγ is the Levi-Civita √ √ symbol (the value of ελαβγ is −g when λαβγ is an even permutation of 0123, is − −g when λαβγ is an odd permutation, and is 0 when λ, α, β, and γ are not all distinct). In other treatments (e.g., Poisson,2 York,42 Gibbons and Hawking43), the boundary terms in the action are expressed in terms of the extrinsic curvature (which is also known as the second fundamental form) of ∂D. However, the second fundamental form is not a well- deﬁned quantity if the hypersurface is null. Consequently, in those formalisms, ∂D cannot

50 be null. Our method, which is nevertheless completely straightforward, has the advantage of allowing for the possibility that ∂D is null. To this end, we introduce boundary symbols. The boundary symbol Bλ does not transform like a 4-vector. It is a coordinate-dependent construct deﬁned by the equation:

λ λσ µν B := g g (gσν,µ − gµν,σ) (A.2)

Note that the boundary symbols are deﬁned in this way so that the principle of least action, when applied to (A.1), leads to the Einstein ﬁeld equation.

We shall split S into parts. There is a cosmological part SΛ: Z 4 √ Λ SΛ := d x −g , (A.3) D 8π a Ricci part SR: Z 4 √ R SR := d x −g , (A.4) D 16π a matter part SM : Z 4 √ SM := d x −g L, (A.5) D a boundary term: Z 1 λ SB = dΣλB , (A.6) 16π ∂D and an additive constant S0, which, being a constant, has zero variation. The total variation δS is then:

δS = δSΛ − δSR + δSB + δSM (A.7)

A.2 The cosmological part

√ 1 √ µν Since δ −g = − 2 −g gµνδg , the variation of the cosmological part is: Z 4 √ Λ δSΛ = δ d x −g D 8π Z 1 4 √ µν = − d x −g Λgµνδg . (A.8) 16π D

51 A.3 The Ricci part

One gets that the variation of the Ricci part is:

Z µν 4 √ g Rµν δSR = δ d x −g D 16π Z 1 4 √ 1 µν µν = d x −g Rµν − Rgµν δg + g δRµν (A.9) 16π D 2

The variation in the Ricci tensor is:

λ λ δRµν = ∇λ δΓ µν − ∇ν δΓ µλ , (A.10)

where the covariant derivatives are calculated with respect to the reference metric gµν. It

λ λ is meaningful to take covariant derivatives of δΓ µν because δΓ µν transforms like a tensor λ even though Γ µν itself is not a tensor. Equation (A.10) is called the Palatini identity (see page 86 of the textbook by Straumann32). The last term in (A.9) involves the integral: Z Z 4 √ µν 4 √ µν λ λ d x −g g δRµν = d x −g g ∇λ δΓ µν − ∇ν δΓ µλ (A.11) D D

Since the metric is covariantly constant, we can write: Z Z 4 √ µν 4 √ µν λ µν λ d x −g g δRµν = d x −g ∇λ g δΓ µν − ∇ν g δΓ µλ (A.12) D D

Relabeling indices, one writes: Z Z 4 √ µν 4 √ µν λ µλ ν d x −g g δRµν = d x −g ∇λ g δΓ µν − ∇λ g δΓ µν (A.13) D D

Using Gauss’ theorem (see page 69 in the book by Poisson2), we can express the right hand side of (A.13) as an integral over ∂D: Z Z 4 √ µν µν λ µλ ν d x −g g δRµν = dΣλ g δΓ µν − g δΓ µν (A.14) D ∂D

The variation of a Christoﬀel symbol is:

1 1 δΓλ = δgλσ (g + g − g ) + gλσ (δg + δg − δg ) (A.15) µν 2 σµ,ν σν,µ µν,σ 2 σµ,ν σν,µ µν,σ 52 Since δgµν = 0 on ∂D, one has:

λ 1 λσ δΓ = g (δgσµ,ν + δgσν,µ − δgµν,σ) (A.16) µν ∂D 2

Using (A.16), we can rewrite (A.14) as: Z Z 4 √ µν µν λσ d x −g g δRµν = dΣλg g (δgσν,µ − δgµν,σ) (A.17) D ∂D

The right hand side of (A.17) can be expressed in terms of δBλ. Taking the variation of (A.2), one gets that:

λ µν λσ λσ µν λσ µν δB = g δg + g δg (gσν,µ − gµν,σ) + g g (δgσν,µ − δgµν,σ) (A.18)

Since δgµν = 0 on ∂D, we have:

λ λσ µν δB ∂D = g g (δgσν,µ − δgµν,σ) (A.19)

Using (A.19), we can rewrite (A.17) as: Z Z 4 √ µν λ d x −g g δRµν = dΣλδB (A.20) D ∂D

So: Z 1 4 √ 1 µν δSR = d x −g Rµν − Rgµν δg 16π D 2 Z 1 λ + dΣλδB (A.21) 16π ∂D A.4 The boundary term

The variation of the boundary term is: Z 1 λ δSB = δ dΣλB 16π ∂D Z α β γ 1 3 ∂x ∂x ∂x λσ µν = δ d y ελαβγ 1 2 3 g g (gσν,µ − gµν,σ) (A.22) 16π ∂D ∂y ∂y ∂y

53 Although the Levi-Civita symbol depends on the metric, we are imposing the condition that δgµν = 0 on ∂D, so the δ-operator can be commuted with most of the factors on the right hand side of (A.22) and one gets:

Z α β γ 1 3 ∂x ∂x ∂x λσ µν δSB = d y ελαβγ 1 2 3 g g (δgσν,µ − δgµν,σ) 16π ∂D ∂y ∂y ∂y Z 1 λ = dΣλδB (A.23) 16π ∂D Note that (A.23) will cancel with the second term in (A.21) upon taking the diﬀerence.

A.5 The matter part

The variation of the matter part is: Z 4 √ δSM := δ d x −g L D Z 4 √ δL 1 µν = d x −g µν − gµνL δg (A.24) D δg 2 A.6 Conclusion

From equations (A.8), (A.21), (A.23), and (A.24), we get that the total variation δS is:

δS = δSΛ − δSR + δSB + δSM Z Z 1 1 µν δL 1 µν = − dv Λgµν + Rµν − Rgµν δg + dv µν − gµνL δg 16π D 2 D δg 2 (A.25)

The principle of least action (δS/δgµν = 0) thus implies:

1 δL R − Rg + Λg = 8π 2 − g L (A.26) µν 2 µν µν δgµν µν

Equation (A.26) is the Einstein field equation, provided that one defines the stress-energy tensor Tµν for the matter-fields as:

δL T := 2 − g L (A.27) µν δgµν µν

54 Note that equation (A.27) disagrees in sign with the corresponding equation on page 125

2 of the textbook by Poisson because Poisson takes the action to be of the form S = SM +

SR + etc, whereas in our present work the action is of the form S = SM − SR + etc.

µν If, in taking the variations of g , we restrict variations of gµν,σ in a certain way on ∂D (e.g., as suggested by Ciufolini and Wheeler44 at the bottom of page 23 of their textbook) then we do not need the boundary terms. For example, if one insisted that δgµν,σ = 0 on ∂D, then by (A.19) one would get that δBλ = 0 on ∂D. Thus, it appears that there are at least two options. One can either restrict variations of the partial derivatives of the metric tensor at the boundary, or one can add the boundary terms. In fact, there are many ways to derive the Einstein ﬁeld equation through variational techniques, and we will not attempt to name them all, but one worth mentioning here is the Palatini method. In the Palatini method (see, e.g., page 25 of the textbook by Ciufolini and Wheeler44), the boundary terms are omitted from the action S and one varies S with

µν λ µν λ respect to both the cometric g and the connection Γ µν, where g and Γ µν are treated as if they were completely independent objects.

55 Appendix B

On the existence and stability of the Ratra-Peebles scalar ﬁeld tracker solution in a curvature-dominated universe

The original ΦCDM model, developed by Ratra and Peebles in 1988, assumed a flat Fried- mann universe (K2 = 0). Recently, the author has been involved in a project1 which aims to generalize the ΦCDM model to non-flat Friedmann cosmologies. In the present Appendix, we will generalize the Ratra-Peebles scalar field tracker solution from the case of a flat universe to a non-flat universe. After setting up the necessary formalism in Section B.1, we will explain what a tracker solution is and establish the existence and properties of such a solution in Sections B.2 and B.3. In Section B.4, we will set up and discuss the problem of determining whether the tracker is stable with respect to small perturbations from the gravity sector in a curvature-dominated universe. We use approximations to argue for stability with respect to the gravitational perturbations; we do not have a mathematically rigorous solution for this latter problem.

1Joint work with Anatoly Pavlov, Bharat Ratra, and Khaled Saaidi. 16 The author is greatly indebted to Anatoly Pavlov for fruitful collaboration regarding the content presented in this Appendix.

56 B.1 Introduction

In 1988, Ratra and Peebles proposed a cosmological model (ΦCDM) based on the following action:23

Z 2 4 √ mp 1 µν S = d x −g −R + g ∂µΦ∂νΦ − V (Φ) + L (B.1) D 16π 2

In (B.1), we are intentionally suppressing the so-called boundary terms discussed in Ap- pendixA, 2 and we shall be following the notation of Peebles and Ratra23 except as noted below. In (B.1), the ﬁeld Φ is the Ratra-Peebles scalar ﬁeld (its potential V (Φ) is given by

3 equation (B.2) below), mp is the Planck mass, L is the Lagrangian density for all matter- ﬁelds except the Φ-ﬁeld, and g, R, etc., are as explained in AppendixA. The potential to be studied is:

κ V (Φ) = m 2Φ−α, (B.2) 2 p where α and κ are (actually interdependent — see equation (6) in Peebles and Ratra23) positive constants. We remark that if we were to follow the original notation of Ratra and Peebles25 completely, we would have put a nonstandard factor of 1/2 in front of V (Φ) in (B.1), and we

2 −α would have written V (Φ) = κmp Φ instead of (B.2). In this respect, and in this respect alone, we are intentionally deviating from the Ratra-Peebles notation. Aside from nonstandard factors of 1/2, another unusual aspect of the Ratra-Peebles formalism (unusual from the perspective of the present author) is that they have a constant

2 factor mp /(16π) multiplying their Lagrangian density for the Φ-ﬁeld. Such a practice deviates in a minor way from the general formalism of Section 2.3.3, but we will not, in the present Appendix, break with this aspect of the original formalism.

2The boundary terms are not always necessary anyway. For example, they are not used in the Palatini method. 3 In fact, mp = 1 according to the preferred units of the present report.

57 Applying appropriate variational principles to (B.1) leads to the ﬁeld equation:

1 8π Rµν − Rgµν = 2 (Tµν + Qµν) , (B.3) 2 mp where Tµν is the stress-energy tensor of the matter-ﬁelds other than Φ, and Qµν is the stress-energy tensor of the Φ-ﬁeld:

m 2 Q = p 2∂ Φ∂ Φ − gζξ∂ Φ∂ Φ − 2V (Φ) g . (B.4) µν 32π µ ν ζ ξ µν

Equation (B.4) can be obtained from (A.27), by taking the Lagrangian density for the scalar

2 µν field as LΦ = mp /(16π) {(1/2)g ∂µΦ∂νΦ − V (Φ)}. Moreover, by applying the principle of least action to LΦ with respect to the field variable Φ, one gets (as in Section 2.3.3) the field equation:

µν 0 ∇µ (g ∂νΦ) + V (Φ) = 0 (B.5)

Specializing to the Friedmann metric (2.1) and assuming that Φ is a function of t only, equation (B.5) becomes:

a˙(t) Φ¨ + 3 Φ˙ + V 0(Φ) = 0, or a(t) a˙(t) Φ¨ + 3 Φ˙ − AΦ−(α+1) = 0, (B.6) a(t)

2 where we use A := καmp /2 for the sake of convenience. Furthermore, specializing to (2.1), the nonzero components of the tensor Qµν become (with respect to the (t, r, θ, ϕ) coordinate basis):

m 2 Q = p Φ˙ 2 + κm 2Φ−α (B.7) tt 32π p ! m 2 Φ˙ 2 − κm 2Φ−α Q = p p a(t)2 (B.8) rr 32π 1 − K2r2 m 2 Q = p Φ˙ 2 − κm 2Φ−α r2a(t)2 (B.9) θθ 32π p m 2 Q = p Φ˙ 2 − κm 2 r2a(t)2 sin2 θ (B.10) ϕϕ 32π p

58 For the Friedmann metric, equation (B.3) yields, in similarity with (2.32):

3˙a(t)2 8π 2 = 2 (ρM + ρR + ρΦ + ρK ) , (B.11) a(t) mp where ρM is the energy density of matter, ρR is the energy density of radiation, ρΦ := 2 ˙ 2 mp /(32π) Φ + 2V (Φ) is the energy density of the Ratra-Peebles Φ-ﬁeld, and ρK := 2 2 2 −(3mp K )/(8πa(t) ) is the eﬀective energy density of curvature.

B.2 A special solution

The concept of a tracker solution to a system of differential equations is similar to the more common concept of an attractor. Both trackers and attractors can be thought of as special solutions that represent what other solutions eventually evolve into under a wide range of initial conditions. The main distinction to be made is that, properly speaking, attractors are fixed points in phase space, but trackers move around in phase space as time progresses. In other words, a tracker is a time-dependent attractor. In the present section, we are going to find a special tracker solution for the Ratra-Peebles Φ-field. We are not going to develop an explicit equation for this tracker solution as a continuous function of time. That would probably be too difficult. Our approach will be similar to that of Ratra and Peebles.25 That is, we will consider discrete epochs where the cosmic scale factor can be approximated by a power law

n a(t) = a0(t − t0) , (B.12) and derive an attractor solution for each epoch. The tracker is then approximated as a discrete frame-by-frame sequence of attractors in phase space. As explained in Section 2.6, the cosmic scale factor can often be approximated by such a power law. For example, when the Universe is radiation-dominated one uses n = 1/2, when the Universe is matter- dominated one uses n = 2/3, for the curvature-dominated case one has n = 1, etcetera. We remind the reader that we are not using a0 to denote the current cosmological epoch as is

59 normally done, and the actual values of the constants a0 and t0, as well as n, will depend on which epoch we are studying. Ratra and Peebles25 have already established the existence of the tracker solution for the radiation and matter-dominated case, but since they were studying ﬂat cosmology they did not need to consider the curvature-dominated case. However, for a negatively curved (spatially hyperbolic) universe, a tracker solution can be found in the curvature-dominated case.4 In fact, we will now demonstrate that the Ratra-Peebles tracker can be obtained, all in one go, for any case where the scale factor is given by (B.12). To this end, substituting (B.12) into (B.6) gives: 3n Φ¨ + Φ˙ − AΦ−(α+1) = 0 (B.13) t − t0

We seek a special solution Φ = Φ0 for (B.13) having the form:

m Φ0 = B(t − t0) , (B.14) where B and m are constants. (This special solution Φ0 will turn out to be our Ratra-Peebles tracker solution.) Making the substitution Φ = Φ0 into (B.13) gives:

m−2 m−2 −(α+1) −m(α+1) m(m − 1)B(t − t0) + 3nmB(t − t0) − AB (t − t0) = 0

(B.15)

By (B.15), we require that:

m − 2 = −m(α + 1) (B.16)

Solving for m gives: 2 m = (B.17) α + 2 Moreover, equation (B.15) also requires that:

m(m − 1)B + 3nmB − AB −(α+1) = 0, (B.18)

4Recall that a curvature-dominated epoch can only happen in a negatively curved (or hyperbolic) cosmology.

60 which yields the following nice relation:

AB−(α+2) = m(m − 1) + 3nm (B.19)

Solving for B gives:

1 A α+2 B = (B.20) m(m − 1) + 3nm We now have:

m Φ0 = B(t − t0) 1 A α+2 2 = (t − t )m, where m = . (B.21) m(m − 1) + 3nm 0 α + 2 It can be verified that this checks out as a solution to (B.13). In the next section, we will show that for fixed n, a0, and t0, our solution Φ0 is an attractor. Since the value of B depends on n, and n changes with each cosmological epoch, we have a different Φ0 from epoch to epoch and so, once it is established that Φ0 behaves like an attractor, we are apt to think of it as a tracker.5

B.3 The attractor property

In order to establish the fact that our special solution Φ0 is an attractor for ﬁxed n, a0, and t0, let us make a change of variables (Φ, t) 7→ (u, τ) where:

Φ(t) = Φ0(t)u(t) (B.22)

τ t = e + t0 (B.23)

5 As mentioned in Section 2.7, the Ratra-Peebles field satisfies the WEC. Note that Φ0 violates the ˙ 2 A −α SEC (and so conforms to our definition of dark energy) if and only if Φ0 < α Φ0 . Moreover, the condition ˙ 2 A −α Φ0 < α Φ0 is equivalent to the condition 2n+(n−1)α > 0. So for the radiation-dominated case (n = 1/2), the field Φ0 violates the SEC if and only if α < 2. For the matter-dominated case (n = 2/3), Φ0 violates the SEC if and only if α < 4. For the curvature-dominated case (n = 1), the field Φ0 violates the SEC for all α. (It is always assumed that α > 0.) Hence, in principle, the value of α could tell us interesting things about the history of the Φ0 field. For example, if α = 3, then the dark energy properties (i.e., SEC violating properties) were yet not present in the Φ0 field during the radiation-dominated epoch but were acquired sometime around the matter-dominated epoch. If α > 4, then the Φ0 field did not acquire its dark energy properties until after the matter-dominated epoch had effectively ended. Note that if Φ0 violates the SEC throughout the whole history of the Universe, then it can be inferred from the above that α . 2 (we have introduced the approximation symbol here because we are dealing with approximations).

61 We want to express the phase-space equations for p (:= u0) and p0 (where primes denote diﬀerentiation with respect to τ). To this end, substituting (B.22) into equation (B.13) gives:

¨ ˙ 3n ˙ 3n −(α+1) −(α+1) Φ0u + 2Φ0u˙ + Φ0u¨ + Φ0u + Φ0u˙ − AΦ0 u = 0 t − t0 t − t0 (B.24)

¨ ˙ −(α+1) Since Φ0 is a particular solution to (B.13), we know that Φ0 + (3n/(t − t0))Φ0 = AΦ0 . So equation (B.24) can be written as:

˙ 3n −(α+1) −(α+1) 2Φ0u˙ + Φ0u¨ + Φ0u˙ + AΦ0 u − u = 0 (B.25) t − t0

m Using Φ0 = B(t − t0) , we write:

2m m m 3n m B(t − t0) u˙ + B(t − t0) u¨ + B(t − t0) u˙ t − t0 t − t0 −(α+1) −m(α+1) −(α+1) +AB (t − t0) u − u = 0

(B.26)

From (B.16), we know that −m(α + 1) = m − 2, so:

2m m m 3n m B(t − t0) u˙ + B(t − t0) u¨ + B(t − t0) u˙ t − t0 t − t0 m −(α+1) (t − t0) −(α+1) +AB 2 u − u = 0 (t − t0) (B.27)

After some algebra, one gets:

2 −(α+2) −(α+1) 2m(t − t0)u ˙ + (t − t0) u¨ + 3n(t − t0)u ˙ + AB u − u = 0

(B.28)

Using (B.19), we can eliminate A and B from (B.28), resulting in:

2 −(α+1) 2m(t − t0)u ˙ + (t − t0) u¨ + 3n(t − t0)u ˙ + (m(m − 1) + 3nm) u − u = 0

(B.29)

62 Let us use primes to denote diﬀerentiation with respect to τ, where τ is deﬁned through

0 2 00 0 (B.23). Note that (t − t0)u ˙ = u and (t − t0) u¨ = u − u , so (B.29) can be rewritten as:

2mu0 + u00 − u0 + 3nu0 + (m(m − 1) + 3nm) u − u−(α+1) = 0 (B.30)

We are now in the position to write down the (u, p) phase-space equations:

u0 = p (B.31)

p0 = (1 − 2m − 3n)p − (m(m − 1) + 3nm) u − u−(α+1) (B.32)

Our special solution Φ0 corresponds to the critical point (u, p) = (1, 0) in phase-space. In fact there do exist other critical points: where u is a complex root of unity, but these are not relevant to the problem at hand because we are dealing with real variables (cf. Ratra and Peebles25). Next, we will take the linearization of the phase-space equations about the critical point. To this end, write:

F (u, p) = p (B.33)

G(u, p) = (1 − 2m − 3n)p − (m(m − 1) + 3nm) u − u−(α+1) , (B.34) and take partial derivatives:

∂uF (u, p) = 0 ∂pF (u, p) = 1 (B.35)

−(α+2) ∂uG(u, p) = − (m(m − 1) + 3nm) 1 + (α + 1)u

∂pG(u, p) = 1 − 2m − 3n (B.36)

Evaluating these partial derivatives at the critical point, we have:

∂uF (1, 0) = 0 ∂pF (1, 0) = 1 (B.37)

∂uG(1, 0) = − (m(m − 1) + 3nm)(α + 2)

∂pG(1, 0) = 1 − 2m − 3n (B.38)

63 So the linearization of the phase-space equations about our critical point can be expressed in matrix form as:

u0 0 1 u − 1 = (B.39) p0 −(α + 2)(m(m − 1) + 3mn) 1 − 2m − 3n p

Using (B.17) to express m in terms of α, we can rewrite the 2 × 2 matrix M appearing in (B.39) as:

0 1 M = 2(α−3nα−6n) α−3nα−6n−2 (B.40) α+2 α+2 One gets that the eigenvalues of the M are:

p λ1,2 = f(α, n) ± i g(α, n), (B.41) where:

(1 − 3n)α − 2(3n + 1) f(α, n) = (B.42) 2(α + 2) 6n(α + 2)(5α + 6) − 9n2(α + 2)2 − (3α + 2)2 g(α, n) = (B.43) 4(α + 2)2

If f(α, n) < 0 and g(α, n) > 0, then the eigenvalues λ1 and λ2 indicate that the critical point is an inwardly spiraling ﬁxed-point attractor in the phase-space. For the curvature-dominated case n = 1, we get that:

−4 − α f(α, 1) = (B.44) α + 2 3α2 + 12α + 8 g(α, 1) = (B.45) (α + 2)2 √ Note that f(α, 1) < 0 and g(α, 1) > 0 for all α ∈ (−∞, −4) ∪ (−2 + 2/ 3, +∞). In particular, f(α, 1) < 0 and g(α, 1) > 0 for all α > 0. For the physical model that we are actually interested in, it is required that α > 0, and so the special solution Φ0 is an attractor for all α-values of interest. Although the radiation and matter-dominated cases were previously studied by Ratra and Peebles,25 let us treat them with brief remarks. For the radiation-dominated case, we

64 have n = 1/2. This gives:

−10 − α f(α, 1/2) = (B.46) 4(α + 2) 15α2 + 108α + 92 g(α, 1/2) = (B.47) 16(α + 2)2

Note that for this case, our critical point is an inwardly spiraling attractor for all α ∈ √ (−∞, −10) ∪ (16 6 − 54)/15, +∞. In particular, it is an attractor for all α > 0. For the matter-dominated case, we have:

−6 − α f(α, 2/3) = (B.48) 2(α + 2) 7α2 + 36α + 28 g(α, 2/3) = , (B.49) 4(α + 2)2 √ and our critical point is an attractor for all α ∈ (−∞, −6) ∪ (8 2 − 18)/7, +∞. In particular, it is an attractor for all α > 0.

B.4 Gravitational back-reaction

The analysis of the previous section did not take into account the gravitational back-reaction. By Einstein’s field equation, a perturbation in the scalar field will induce perturbations in the spacetime geometry, and vice versa. In the present section, we shall set up and discuss the problem of dealing with the gravitational back-reaction associated with long-wavelength inhomogeneities in the scalar field. The conjecture is that the tracker solution Φ0 remains stable with respect to such perturbations, but this problem leads to a system of differential equations which is not as straightforward to analyze as the system studied in the previous section. We will present evidence of stability using an approximation argument which was developed in a collaboration between the present author and Anatoly Pavlov. Unfortunately, a rigorous mathematical proof is not currently available. We are going to study spacetime perturbations about a curved Friedmann background

65 metric. To this end, we start by writing:

2 µ ν ds =g ˜µνdx dx

µ ν = (gµν + δgµν)dx dx , (B.50)

where gµν is the unperturbed homogenous Friedmann background and δgµν is a nonhomoge- nous perturbation. We will work in terms of the Friedmann coordinates (t, r, θ, ϕ). These coordinates are time-orthogonal so we can use the so-called synchronous gauge (see e.g., Ratra and Peebles,45 or Chapter V of the 1980 book6 by Peebles46). With the synchronous gauge, we write:  1 0 0 0  a(t)2  0 − 1−K2r2 0 0  gµν =   (B.51)  0 0 −a(t)2r2 0  0 0 0 −a(t)2r2 sin2(θ) and:  0 0 0 0  hrr 2  0 1−K2r2 hrθ hrϕ  δgµν = εa(t)  2  , (B.52)  0 hrθ r hθθ hθϕ  2 2 0 hrϕ hθϕ r sin (θ)hϕϕ

2 where ε ∼ 0 and each hij is a function of t, r, θ, and ϕ. The equation of motion for the Φ-ﬁeld in a spacetime with cometricg ˜µν, reads:

˜ µν 0 ∇µ(˜g ∂νΦ) + V (Φ) = 0, (B.53) where we put the tilde over ∇ to emphasize that the connection is the one determined by

25 the perturbed metricg ˜µν. Note that the Ratra and Peebles have a (nonstandard) factor of 1/2 in front of V 0(Φ), which we are intentionally getting rid of. Henceforth we will use φ to denote a long-wavelength perturbation in the Φ-ﬁeld. The perturbed scalar ﬁeld is thereby written:

µ µ Φ(x ) = Φ0(t) + φ(x ), (B.54)

6Though the synchronous gauge is discussed somewhat in Peebles’s 1980 book, 46 the name ‘synchronous gauge’ is not mentioned in that reference.

66 where Φ and φ are spacetime dependent, but Φ0, which depends only on t, is a solution to the scalar ﬁeld equation of motion in the unperturbed homogenous Friedmann background. Thus:

µν 0 ∇µ(g ∂νΦ0) + V (Φ0) = 0, or a˙ καm2 Φ¨ + 3 Φ˙ − p Φ−(α+1) = 0 (B.55) 0 a 0 2 0

Plugging (B.54) into (B.53) gives, to ﬁrst-order in φ:

˜ µν 0 0 = ∇µ(˜g ∂ν (Φ0 + φ)) + V (Φ0 + φ)

˜ µν ˜ µν 0 00 = ∇µ(˜g ∂ν (Φ0)) + ∇µ(˜g ∂ν (φ)) + V (Φ0) + V (Φ0)φ (B.56)

In fact, after some algebra and tensor calculus, equation (B.56) can be written as:

3˙a 1 1 φ¨ + φ˙ − ∇2φ + V 00(Φ )φ − h˙ Φ˙ = 0, (B.57) a a2 0 2 0

1 µν 2 where h = hrr + hθθ + hϕϕ = − ε (g δgµν), and ∇ is the Laplacian for the 3-dimensional homogenous and isotropic space of uniform curvature K2. In terms of the coordinates that we are presently using, we have:

1 ∂ ∂ ∂ 1 ∂ ∂ 1 ∂2 ∇2 = (r2 − K2r4) + K2r + sin(θ) + r2 ∂r ∂r ∂r r2 sin(θ) ∂θ ∂θ r2 sin2(θ) ∂ϕ2 (B.58)

For the case K2 = 0, equation (B.58) corresponds to the usual 3-dimensional ﬂat-space Laplacian expressed in spherical coordinates. We note that equation (B.57) corresponds to equation (3.11) in the 1995 paper by Ratra and Peebles45 dealing with inﬂation theory in a negatively curved (hyperbolic) universe, provided that one accounts for the fact that we are eschewing the 1/2 factor that Ratra and Peebles habitually wrote in front of the potential V .

The stress-energy tensor Qµν for the (perturbed) ﬁeld Φ (= Φ0 + φ) has the form: m 2 Q = p 2∂ Φ∂ Φ − g˜ζξ∂ Φ∂ Φ − 2V (Φ) g˜ (B.59) µν 32π µ ν ζ ξ µν 67 µν One gets that, to ﬁrst-order, the trace Q =g ˜ Qµν is:

m2 m2 Q = p 4V (Φ ) − Φ˙ 2 + p 2φV 0(Φ ) − Φ˙ φ˙ ε (B.60) 16π 0 0 8π 0 0

We shall require, to ﬁrst-order, the Qtt-component of the stress-energy tensor:

m2 m2 Q = p Φ˙ 2 + 2V (Φ ) + p Φ˙ φ˙ + V 0(Φ )φ ε (B.61) tt 32π 0 0 16π 0 0

As for the Ricci tensor Rµν, we will require the Rtt-component. To ﬁrst-order, it is:

3¨a a˙ 1 R = − + h˙ + h¨ ε (B.62) tt a a 2

By comparing the tt-components on both sides of (1.35) one gets that:7

2˙a h¨ + h˙ = 2Φ˙ φ˙ − V 0(Φ )φ (B.63) a 0 0

Equation (B.63) corresponds to equation (3.14) previously derived by Ratra and Peebles45 in 1995, modulo the factors of 1/2 that we are dropping intentionally.

Since we are interested in the curvature-dominated epoch, let us take a = a0 · (t − t0). Then equations (B.57) and (B.63) become:

2 ¨ 3 ˙ ∇ φ 00 1 ˙ ˙ φ + φ − 2 2 + V (Φ0)φ = hΦ0 (B.64) t − t0 a0(t − t0) 2

¨ 2 ˙ ˙ ˙ 0 h + h = 2Φ0φ − V (Φ0)φ, (B.65) t − t0

Following Ratra and Peebles,45 we transform equation (B.64) from (3-dimensional) position space to (3-dimensional) momentum space. Under this transformation, one has ∇2φ = L2φ, where L2 is a scalar (L2 is not necessarily positive). In accordance with the 1995 paper by Ratra and Peebles,45 in the negative curvature case, one has L2 ≤ −1 with L2 → −1 for long-wavelength perturbations and L2 → −∞ for short-wavelength perturbations. In

7 We are assuming that the perturbations φ and δgµν do not signiﬁcantly aﬀect matter and radiation.

68 the present study, we are interested in long-wavelength perturbations.8 On transforming to (3-dimensional) momentum space, we can rewrite (B.64) as:

2 ¨ 3 ˙ L 00 1 ˙ ˙ φ + φ − 2 2 φ + V (Φ0)φ = hΦ0 (B.66) t − t0 a0(t − t0) 2

0 00 We will need V (Φ0) and V (Φ0). These are:

0 −(α+1) V (Φ0) = −AΦ0 (B.67) and

00 −(α+2) V (Φ0) = (α + 1)AΦ0 (B.68)

Thus, transforming (B.65) into (3-dimensional) momentum space,9 and pairing it with equation (B.66) gives the system:

2 ¨ 3 ˙ L −(α+2) 1 ˙ ˙ φ + φ − 2 2 φ + (α + 1)AΦ0 φ = hΦ0 (B.69) t − t0 a0(t − t0) 2

¨ 2 ˙ ˙ ˙ −(α+1) h + h = 2Φ0φ + AΦ0 φ (B.70) t − t0

Deﬁning J by:

2 2 L J := (α + 1)(m + 2m) − 2 , (B.71) a0

m and substituting Φ0 = B(t − t0) into (B.69) and (B.70), leads to:

¨ 3 ˙ J mB ˙ m−1 φ + φ + 2 φ = h(t − t0) (B.72) t − t0 (t − t0) 2

¨ 2 ˙ m−1 ˙ −(α+1) m−2 h + h = 2mB(t − t0) φ + AB (t − t0) φ, (B.73) t − t0

2 2 We note that since L is negative, and since α, m, and a0 are positive, it follows that J > 0. 8 For the long-wavelength case, φ can be approximated by a power-law in t, but for the short-wavelength case, one can use the WKB approximation (which is not studied in the present work). 9Since equation (B.65) does not involve spatial derivatives, it still looks the same after we transform from position space to momentum space.

69 −2 For the curvature-dominated case, one has ρK ∼ (t − t0) and:

2m−2 ρΦ (t − t0) 2m ∼ −2 = (t − t0) (B.74) ρK (t − t0)

m p m Thus, (t − t0) ∼ ρΦ/ρK in the curvature-dominated case (or in other words (t − t0) = p C ρΦ/ρK for some constant C), and equations (B.72)-(B.73) can be written as: ˙ r ¨ 3 ˙ J mBC h ρΦ φ + φ + 2 φ = (B.75) t − t0 (t − t0) 2 t − t0 ρK

r −(α+1) r ¨ 2 ˙ 2mBC ρΦ ˙ AB C ρΦ h + h = φ + 2 φ (B.76) t − t0 t − t0 ρK (t − t0) ρK

We will now search for approximate solutions to these equations using techniques from p perturbation theory. Since we are in the curvature-dominated case, where ρΦ/ρK is small, we begin by searching for approximate solutions to (B.75) and (B.76) where the source terms on the right hand side are neglected. That is, we ﬁrst solve the following equations to get a ﬁrst approximation for φ and h˙ :

¨ 3 ˙ J φ + φ + 2 φ = 0 (B.77) t − t0 (t − t0)

2 h¨ + h˙ = 0 (B.78) t − t0

Once we have these first approximations, we will plug them back into equations (B.72) and (B.73) in order to obtain new differential equations which can then used to derive a second approximation which includes correction terms. We will then look at these second approximations, which will turn out to be equivalent to our first approximations plus correction p terms of order ρΦ/ρK , and we will see that the second approximations indicate that the perturbations φ and h˙ rapidly die out. (At best, this can only amount to an approximation argument, which may provide evidence of stability, but not a rigorous mathematical proof.)

70 The general solution to (B.77) is:

 C C 1 √ + 2 √ if J < 1  1+ 1−J 1− 1−J  (t−t0) (t−t0)   C1 + C2 ln(t−t0) if J = 1 φ(t) = t−t0 t−t0 (B.79)   √ √  C1 cos( J−1 ln(t−t0)) C2 sin( J−1 ln(t−t0))  + if J > 1, t−t0 t−t0 where C1 and C2 are constants of integration. The general solution to (B.78) is:

˙ −2 h = C0(t − t0) , (B.80)

where C0 is a constant of integration. Plugging (B.80) into equation (B.72) and solving the resulting diﬀerential equation for φ gives a new approximation for φ, which now includes correction terms:

 C (t−t )m C1 √ C2 √ 3 0 + + 2 if J < 1  (t−t )1+ 1−J (t−t )1− 1−J (m +J−1)(t−t0)  0 0   m  C1 C2 ln(t−t0) C3(t−t0) + + 2 if J = 1 φ(t) = t−t0 t−t0 m (t−t0) (B.81)   √ √  C cos J−1 ln(t−t ) C sin J−1 ln(t−t ) m  1 ( 0 ) 2 ( 0 ) C3(t−t0)  + + 2 if J > 1, t−t0 t−t0 (m +J−1)(t−t0)

p m where C3 := mBC0/2. Since ρΦ/ρK ∼ (t − t0) , the correction terms C3 are of order p ρΦ/ρK times a decreasing function of t (decreasing for t > t0). Since we are in the curvature-dominated case, these correction terms must be very small. So from (B.81), which is our second approximation for φ, it appears that the perturbations φ are rapidly dying out. Plugging (B.79) into equation (B.73) and solving the resulting diﬀerential equation for

71 h˙ gives a new approximation for h˙ , which now includes correction terms: h˙ (t) =

√ √  α+2 m α+2 m C C1{2mB (1+ 1−J)−A}(t−t0) C2{2mB (1− 1−J)−A}(t−t0)  0 + √ √ − √ √ if J < 1  (t−t )2 α+1 2+ 1−J α+1 2− 1−J  0 B ( 1−J−m)(t−t0) B ( 1−J+m)(t−t0)     (t−t )m A(mC −C )+2mBα+2(C −mC +mC )+mC (A−2mBα+2) ln(t−t )  C0 0 { 1 2 2 1 2 2 0 }  2 + 2 α+1 2 if J = 1 (t−t0) m B (t−t0)

 √ √  m √ α+2  (t−t0) {[A(mC1−C2 g−1)−2mB (C1(1−J+m)−C2(1+m) J−1)] cos( J−1 ln(t−t0))}  C0  (t−t )2 + Bα+1(m2+J−1)(t−t )2  0 √ 0 √ √  m α+2  (t−t0) {[A(mC2+C1 g−1)−2mB (C2(1−J+m)+C1(1+m) J−1)] sin( J−1 ln(t−t0))}  + α+1 2 2 if J > 1, B (m +J−1)(t−t0) ˙ where C1 and C2 are the same constants appearing in (B.81). Again, the perturbations (h)

p m are dying out over time, assuming ρΦ/ρK ∼ (t − t0) is small.

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